Mawaru Penguindrum/WMG

Princess of the Crystal/SeizonSenryaku!Himari is Momoka.
Seeing how she's guiding the Takakura family,. And also because their facial expressions are really similar.

Himari is dying and her brain is struggling to put it all in context.
Her brain's losing oxygen and coming up with outlandish scenarios in order to convince itself it won't die. That explains the hallucinogenic henshin sequences, the crazy invisible penguins, and why everything, in the end, revolves around her and a search for a MacGuffin that in all likelihood doesn't exist.
 * Unlikely. From all these CAT scant in a doctor's office Himari has a heart condition, and they don't usually work that way.

Kanba, Shouma, and Ringo are all Momoka.
There's the obvious bit about all of them being born on the day Momoka died, but think about it a bit more. Kanba has red hair, and Shouma has blue hair. What happens when you mix red and blue? You get purple and/or magenta, the color of Momoka's hair. Now, brown is considered to be the "normal" color. So Ringo is in essence what is left of Momoka when one removes the "red" and "blue" essence from her.

I say episode ten and onwards begin to pull together the threads, and the Noruniru theme will begin to take on new meaning.
The 'fate diary,' or Penguindrum, is in fact a book from the pocket dimension of the library annex, and it being brought into the mortal universe and being allowed to influence the timeline could cause a dimensional collapse of some sort. The Penguingun lady will be revealed as actually being a bait-and-switch antagonist, whose real motive is either of little significance or some sort of 'rage against god' towards the inhabitants of the annex, who may in fact write fate rather than simply record it, and who are somehow malevolent or with a plan that doesn't take human suffering into consideration. Penguingun will also probably have some sort of fate diary, or know what's recorded in one, and is using it to try to fight against a future with a negative outcome.

And Sanetoshi is actually either the soul of a dead, very camp librarian, or God.

The train mascot characters are Kanba and Shoma.
At some point their memories were altered and their genders swapped. If they ever were truly fictional in-story, they've since become Refugees From TV Land. Further elaboration will require some information on just where those girls come from.
 * Seemingly Jossed by episode 5's flashbacks.
 * They have a higher chance to be the Kahirasei-jins from Utena, given their exactly same function and motifs.
 * Jossed The mascots are the girls seen in the ending theme, Himari's childhood friends Hibari and Hikari, now an idol group called WH (double H).

The hat would take form as Himari's Twin if they give the Penguin Drum.
Just that.

The pink-haired woman in the opening credits was Tenjou Utena, or at least a very strong expy of her
She may also turn into a tribute to the late Tomoko Kawakami, who rumour has it had expressed interest in working with Ikuhara again when she heard he was working on a new series.
 * Jossed Neither a woman, nor Utena. Episode 9 reveals him to be Sanetoshi the librarian.

The penguin hat is a dying species.
Hence, the Survival Strategy. Extending this, the penguindrum is some sort of device which facilitates reproduction, or something that will allow them to contact more of their own species.

Himari will be dead again by the end of the series, this time for good.
Before then, she will deliver a What the Hell, Hero? speech to her brothers regarding some morally questionable things they will have done to prolong her life a few months, having found out about all the Penguindrum business. She'll probably forgive them for it before dying, though -- she doesn't seem vindictive.
 * It's too early to say since there isn't much focus on Himari but at least the Hat can possess her easily so she probably doesn't fight against the possession and by extension the prolonging of her life. And some people have noted that her theme seems to be escapism. Those who want to escape have something they disagree with. In her case, maybe her illness?

The Penguindrum is actually Yuri's spin-dryer that will start the Survival Strategy and will change the fate of all others.
The diary is a Red Herring literally...

Sanetoshi is a hermaphrodite and his helpers are the anthropomorphic black Bunnies of doom.
Or he just be could be a Gender Flipped post-corrupted Utena... (is silenced by the Shadow-Girls)

The Anti-Transformation Transformation Sequence is actually a Journey to The Centre of The Mind for everyone.
Shoma, Kanba and Ringo are transported into Himari's mind while she's being possessed by the Penguin Hat, then she gives A World Of Cardboard speech...

Ringo's supposedly dead sister is the one that keeps shooting all of Kanba's ex-girlfriends.
The reason is unknown for now, but as of episode 6, it seems she knows about Project M, which is apparently Ringo's final plan to get close to Tabuki. She also has her own penguin, so she might also know about the Penguin Drum.
 * There's a good chance that her name is an alias, even if this isn't true.
 * Jossed by episode 15, which features an extended flashback sequence revealing  Furthermore, Episode 16 explains Masako's backstory.

Himari is adopted...
The flashback scene where she shares an apple with a boy takes place at an orphanage.
 * As of episode 19...

The blond boy seen at the beginning of episode 9 is the boy Himari shares an apple with in the orphanage(??) in the flashback in episode 9.

 * Jossed as of episode 19.

That faceless boy, from episode one who interrupts Shoma on his walk home is Himari's soulmate.
Okay this is a little insane and he may have just been thrown in there to give Ringo an early cameo but! Hear me out. In episode 9 while most of his face was obscured, there is a faint outline of grey hair. And this boy in ep 1 has grey-ish hair. Also, people who aren't important and don't matter a great deal to the plot are usually just animated as those odd stick people. He wasn't, and he knew Kanba. And they didn't show us his face, that whole scene and we didn't see it once. Why not? And last, in the opening there is a boy in the same uniform as Kanba and Shoma but is all black. We can't know for sure who he is. So I'm thinking he's Himari's soulmate, or if not a character who will show up later.
 * Turns out, he's actually the Penguin King, Mario. Jossed?
 * Double Jossed.

Yuri is actually Ringo's dead sister Momoka
Who either never really died or was revived. Yuri actually is around the age that Momoka would've been if she was still alive, Tabuki refers to her as a "childhood friend" just like Momoka was. ..
 * With episode 12, this theory may...
 * Definitely
 * Definitely

Momoka died in the 1995 Sarin gas attack on the Tokyo subway.
According to the wiki Ringo's birthday is March 20, 1995, the same day of the attacks. It fits within the timeline and with the theme of fate, only 13 people died in the attack. What were the chances it was Momoka? Also the number 95 and trains are motifs that pop up a lot in the series.

Himari's possessor is Momoka.
Unfortunately I can't remember why this felt like the most brilliant idea ever 3 A.M. last night. All I remember is that it felt like a good explanation to why she is so utterly pissed at Ringo; who would want someone imitate every aspect of their life through a stolen diary, especially a younger sibling who should be living her own life instead of pining for the sister she never knew.

Shouma is adopted
It's odd that in episode 12, when Kenzan-papa is "working", he's only told "it's a boy!" instead of "two boys!". It would also explain why Shouma referred to Himari as "your sister" when calling Kanba out in the first episode and then correcting himself, why Himari's possessor only used Kanba's life energy in the survival strategy and Shouma is always thrown down the chute (she can't use Shouma's energy because he's not as directly related to Himari as much as Kanba is). Therefore, Shouma isn't Kanba's twin brother, but a boy around his same age (maybe the child of ?) raised alongside him and Himari.
 * But why Shouma is practically identical to Kanba, if this is indeed the case?
 * Hm... While Shouma does resemble Kanba, they're not identical: to start, Kanba has rather narrow eyes when compared to Shouma's rounder ones, and then we have the different hair colors (Considering that they're presented as fraternal twins, this makes sense). Also, while Kanba is basically a younger and redheaded Kenzan and Himari is a teenaged Chiemi, Shouma doesn't bear such strong physical resemblances to the parents (him having Chiemi's personality is another story). And if they're two unrelated children about the same age, it could handwaved by presenting them as fraternal twins, since fraternals don't have to look the same. It's still very heavy speculation, but not completely impossible... we'll have to see.
 * Another possibility is that Shouma is  Which
 * As of episode 20?
 * And then episode 23 states that
 * And then episode 23 states that

Shouma has a hidden character flaw related to gluttony.
He may be a glutton for punishment, or something else less conventional, but he has some secret obsession involving consuming something. This idea is ofcourse based on the fact that all the other penguins in the show somehow mirror their owners' personalities: #1 is Lovable Sex Maniac, mirroring Kanba's womanizing ways, #3 is a strong and supportive Yamato Nadeshiko like Himari and Esmeralda is an obsessive stalker always bent on molesting #1 at any given opportunity, mirroring Masako's Yandere nature. So #2 has to have something similar going on.
 * It's also speculated by some that he might have a strong desire to kill or harm either himself or others, covered under his Yamato Nadeshiko traits. He once pins Kanba to the floor,  and has a very erratic behavior when , while #2 keeps spraying bugs everywhere and passes up saving Ringo from drowning. His inner monologues in the novels also squarely put him as a Type B Stepford Smiler: he looks and acts normal, but feels very empty inside, and maybe that's why #2 has... erratic behavior patterns as well, dominated by a possible need to "fill" himself whether with food or with bug killing.

Sanetoshi is the Big Bad
I'm sure this is pretty obvious by now, what with the show's Utena influences, him referring to himself as the "Destination of Fate" (way too similar to "End of the World"), and him appearing with the dark bunnies. And the survival strategies, etc. seem like an obvious stand-in for the duels and the Rose Bride.

Sanetoshi is using the Survival Strategies to revive his lover.
The penguin hats are similar to horcruxes; they contain a portion of the princess's soul. Sanetoshi is using the Strategies to determine which body would be ideal to place the full soul in and thereby revive the princess. However, the princess doesn't love him back, which will cause her to become a Phlebotinum Rebel when she is finally revived (probably in Himari, because seriously, who else?).

The person Satenoshi is talking about in episode 13 might be Momoka
He tells that 16-years ago he met someone like him, but that person refused him. Really, the only female character alive to that point that could fit in is Momoka.
 * All but explicitly confirmed now. And with Ikuni the Shrug of God is the only option.

The scars on Tabuki's fingers...
Were caused by  Wasn't Tabuki's first love Momoka? And with Episode 14 revealing
 * The problem with that is
 * Were they? I only remember them being shown, not before. I could be mistaken.
 * Yep, they are scarred in his childhood scenes as well.

Yamashita will be important somehow
It's thanks to him that Shouma talks to Ringo first, then he takes Shouma to the same spring here  and it's thanks to him   that. Also, some people think that
 * Jossed,

Momoka died on the subway to fix a problem of Tabuki's
Said problem is related to his finger scars.
 * Jossed in Episode 18.

Yuri is a
In episode 14 it is revealed that This seems to suggest that there is something about her

Notice how her father keeps talking about how she was and that he would. His "coup de gras" or whatever he called it in the show would evidently be which under these circumstances would probably lead to, which is where Momoka steps in.

Also, episode 15 opens with Yuri saying: This is continually referred to by her and may be a metaphor for.

Her being also adds some weight to this theory, though it is by no means proof.
 * But if all this is indeed the case, why does she develop injuries to her arm and leg? Wouldn't her father focus on  right away? I find it more probable that Yuri's father was referring to psychological rather than physical "ugliness", and that from his perspective a "work of art" that he created was ugly because it wasn't entirely under his control. I believe that he was aiming to twist Yuri's psyche to complete dependence towards himself with the abuse, and when he realized he couldn't get it work forever, decided to kill and possibly preserve her in some way to permanently make her completely his's.
 * The second paragraph of this post explains it. He could have possibly used the skin from her arm, leg, and face to construct that organ.
 * However, there's how Shouma aka Mr. I "Flinch When I See A Girl's Panties" didn't say anything about Yuri's naked body when he got to see her fully naked. If, he would've pointed it out. Same goes to Ringo; she would've noticed   if she had it, when they were bathing.
 * If anything, the explanation behind that link disproves the idea.

The scars on Tabuki's hands are a result of him using the destiny diary at some point.
Momoka got a scar on her hands (as shown by the bandage in episode 15) for making small-scale alterations to fate. If it is the diary itself (and not Momoka) that warps reality, it stands to reason that other people with the understanding how could use it, such as if Momoka explained it to him. That also further explains why all the factions are after it.

The series and the novels follow different paths of destiny.
They are superficially identical, but at some point there will be a some major event involving the Destiny Diary that will take the two into wildly different directions. And both will at some point give a glimpse to the other reality for the characters.

Momoka died
This would be a believable thing for Momoka to do

The Penguindrum is a drum with a penguin pattern on it.
The closing shot of the first flashback in episode 9 is of a drum with a penguin pattern next to an ocarina. The camera stayed a little too long on this shot for it to be meaningless, also it's literally a Penguin-drum.
 * On the other hand at one point Japanese fans commented that it sounds like a washing machine brand, and we've also zoomed onto a washing machine commercial with a prominent penguin logo. Ikuhara be trolling.
 * If you think about it again though, this might actually be the case.  Most likely it's just trolling though as the troper above has pointed out.

Sanetoshi's ampules are Powered By a Forsaken Child.
It's pretty strongly implied that the Princess of the Crystal extended Himari's life using Kanba's life force, and that she refused to repeat the process because it would kill him. So what do the life-extending ampules used by Dr. Sanetoshi contain? The concentrated life force of everybody who died in the subway bombings 16 years back, and in fact the sole purpose for these terrorist attacks was to harvest life for those who destiny had given a short straw, but deserved a future by the Penguin Faction's judgement.
 * Maybe the child broiler is actually a real place and he uses the crushed souls of all those forsaken children to power his ampules.

Yamashita is Tabuki's little brother.
They both have grey-ish hair, and his being The Faceless might be to hide the family resemblance.

Whatever the Penguindrum is, it will manifest itself as a giant, penguin-shaped mecha.
Those bear-shaped mecha-thingies are supposed to be called "Teddydrums". Do the math.

Shouma's apathy and impotence are the result of him giving a part of his own fate to Himari in the Child Broiler.
There does seem to be an idea going on in the show that there's only so much fate to go around that can't be created or destroyed, but can be altered and moved around.

Kanba is the only true child of the Takakuras.
From what we've seen
 * Jossed as of episode 21.

Those faceless, genderless, bathroom sign-style pedestrians we see every episode are actually came from the Child Broiler.
We saw how many of this children are there, if they don't kill or Ret Gone them, then it will really be enough of them to be that faceless mass we see every episode.
 * I initially thought this as well, but the children look like that before going through the Broiler, so I'm not sure.

Himari is going to do something horrible to Ringo.
Anyone else found her dialogue in making the "wholesome apple curry" kind of ominous? She did seem slightly jealous of Ringo in episode 19, and now she has discovered a very good reason for it.
 * Jossed as of episode 21.

The Princess of the Crystal is Momoka, but doesn't remember.
She knows she has to get the Penguindrum (whatever it is), but doesn't know why. When Himari first sees the diary, the Princess realizes -- without knowing why -- that if she gets the diary, she won't need the Penguindrum, because she'll be able to prevent the current situation from ever having developed. (And if she can prevent the bombings altogether, that means she lives, her family doesn't break apart, and her friends and her baby sister don't become as screwed-up.) Her haughty behavior may or may not be a Jerkass Facade; if it's not, and she does remember her identity, then Yuri and Tabuki become very unreliable narrators wrt her character.

The Princess's outfit is a "bridal" riff on components of Himari's subconscious.
The main body is based on the Double H outfits, and there's a scarf because of her own history with them (see episode 20).

The screens on the subway can tell when nobody's paying attention to them.
This is when they show some of the stranger Double H PSAs, as a way of having fun at the protagonists' expense.

In the last episode, the timeline will be changed one last time.
The penguin logos will be gone, and we'll find out that the subway attack still happened -- but it was done by a different group, and with Sarin gas instead of bombs.

Kenzan and Chiemi Takakura had no children.
Himari and Kanba have already been revealed to not be blood related to the Takakuras, but then comes episode 22. After the credits  Because Shoma is not their son,.
 * The logo on the cage is not the Kiga Group logo. Kiga's logo is divided into black and white halfs. It also isn't the logo of the Penguin Force that is all black with distinct eyebrows. It's the last remaining unexplained penguin logo that was also seen on the watch of the murdered journalist that may represent the enemies of the Kiga Group.
 * That logo has been semi-explained as the logo for 'Pingroup'. Pingroup seem to be pretty large in the Penguindrum universe, almost every product is made by them and even the hospital has their logo everywhere. It's likely that Pingroup control the world and run the Child Broiler.
 * What gets me is the fact that the three penguins were delivered in a box with a Pingroup logo on it. Just who are they loyal to?
 * Double H?
 * Jossed. Episode 12 has Kenzan being informed of the birth of their child.

This is set in the same multiverse as Umineko no Naku Koro Ni
Think about it.

The Men in Black of the Kiga Group are the "invisible children" who come out from the Child Broiler.
They have lost their individuality and their place in the world, and they want to reclaim them by destroying the established order.

Momoka is the reason there are so many penguin-themed product names.
Penguins were her favorite animal, and she discovered her diary's powers while thinking about them.

The Child Broiler is a metaphor for child suicide
Both Tabuki and Himari were despairing over being unwanted by their families and had given up on life. They were saved when someone reached out to them and showed them love, giving them hope. With this show's heavy themes, it wouldn't be surprising for the characters to have been Driven to Suicide at such a young age.