Raised as the Opposite Gender

A couple has a baby that's the opposite gender from the one they wanted or needed. For whatever reason, they decide that rather than keep trying, they're going to raise the existing child as if the child were the desired gender.

Once the kid is a little older, this usually leads to conflict between the child and the parents and/or inner conflict for the child, who has to weigh the desire to be themself against their desire to please their parents (or keep their inheritance or what have you). In a few cases, though, the child turns out to identify just fine with the gender they were raised to be. When the child's gender identity does match their physical sex, they may still retain some Wholesome Crossdresser tendencies.

Can result in a lot of Angst if the child suppresses his/her desire to be him/herself (until it can't be suppressed anymore, like when the child begins to show prominent secondary sexual characteristics, or when the child begins to love someone of their opposite sex), or when the child rebels against the parent's desire and the parent simply can't accept the child for who he/she is.

Beware of inherent spoilers in the examples.

Examples of Raised as the Opposite Gender include:

Girls (/Female-Bodied People) Raised as Boys

Akito in Fruits Basket, due to her Complete Monster mother's irrational jealousy towards her that made he demand for Akito to be raised as a boy. Akito herself has always known she's a girl, but almost everyone except the oldest members of the Souma clan (and later Tohru) ignore it, until she willingly reveals it after her Heel Face Turn.
Tsukasa Shuro in Ayashi no Ceres, according to the tie-in novel focusing on her backstory, didn't even realize she was a girl until she was about 8 years old. She was raised as a boy to protect her from the curse of the Hagoromo, since she's one of the Celestial Maidens.
Lady Oscar in Rose of Versailles. But by the time she meets Andre at age 7, she already knows she is a girl. By the time Oscar joins the military at age 14, most people can tell she is female.
In the Oz series, Princess Ozma was magically disguised as a boy for most of her childhood. Even she didn't know the truth. She seemed to adjust to being a girl very easily, however.
One of the stories from Arabian Nights had a groom reveal to the bride on their wedding night that he was actually a woman raised as a man due to her father putting pressure on her mother for a son.
Tamir (a.k.a. Prince Tobin) from Lynn Flewelling's Tamir Triad was magically transformed into a boy at birth (long story short, the people currently in power required a male heir, but certain groups wanted a return to the country's old matriarchal ways). She didn't know she was a girl until puberty hit and weakened the spell, and had some awkwardness adjusting to being female later.
Gintama's Kyuubei was raised as a boy so that she could succeed her family's clan.
Type Moon seems to like this — Saber from Fate/stay night is a standard "needed a male heir" example, and Shiki from Kara no Kyoukai: may also count, though it's honestly a lot more complicated than that.
Princess Sapphire from The Princess Knight - again raised as a boy due to a law that only male heirs could rule, and further complicated by the fact that she had the "hearts" of both a boy and girl due to a mistake in heaven.
In The Metamorphoses, Iphis's mother raises her daughter as a boy on divine orders to avoid exposing her at birth. Iphis falls in love with another girl named Ianthe, so the goddess Isis turned Iphis into a man and the two got married.
Features in Mark Twain's A Medieval Romance.
Kellen in The Dream-Maker's Magic was raised as a boy by her mother, who was convinced she had actually given birth to a boy and couldn't handle having a daughter instead. Kellen started occasionally dressing as a woman around the time she hit puberty. When she left to go to Wodenderry, she switched back to pretending to be a man for her own safety, but was eventually forced to admit the truth to her friends. It turned out her mother really had given birth to a boy, and the babies were Switched At Birth.
Penny in The Big Bang Theory
Robin "Charles" Scherbatsky in How I Met Your Mother
Celeste in the Gemini books by V. C. Andrews.
"Prince" Igam in Hikari Sentai Maskman
The narrator, Frank, in The Wasp Factory as it turns out her father wanted to experiment and see if it would work.
Genki Saotome in Getter Robo Armaggedon. When we meet her as an adult, she actually identifies as a female, and has become the Action Girl Kei Kuruma. Brilliant plot twist, or an Ass Pull for the ages? Who knows.
Chiyomaru/Kiyone in Genzo. Princess Kiku is a variation: while her gender wasn't concealed at all, her father raised her like a boy and a warrior in order to defeat Yasuke and the Yamiganemaru.
Shintarou Inoue in Sumomomo Momomo, and it's not entirely clear if this happened because the head of the clans that the Inoue clan were aligned with had to be male to fit into an Arranged Marriage, if it's because Shintarou's dad is an Unknown Rival to the main character's dad, and Shintarou's dad wants revenge by proxy, or if it's because the character's dad is just a bit crazy.

Boys (/Male-Bodied People) Raised as Girls

Daisy in the Christopher Durang play Baby with the Bathwater.
The singer in The Who's song "I'm a Boy":

I'm a boy, I'm a boy, but my ma won't admit it; I'm a boy, I'm a boy, but if I say I am, I get it.

Angela in Sleepaway Camp
Lord Fanny in The Invisibles is born as a boy into a family of powerful female witches, and s/he is raised as a girl so that s/he can continue the family tradition (which is forbidden to men). Her/his grandmother even chastises her/his father for producing a male child instead of the preferred female one. Fanny quite easily accepts the gender she is given and grows up to be a transgendered woman with magical powers.
An episode of Law and Order SVU featured a pair of fraternal twins. However, you find out later that they were born identical, and the sister was actually a boy who was given a sex change after a botched circumcision. This was Ripped from the Headlines, oddly enough.
On CSI, The Freudian Excuse of a serial killer is revealed to be a gender identity issue: His father raised him as a boy, his mother as a girl.
On 30 Rock, it's mentioned as a Noodle Incident that Josh was raised as a girl for 10 years.
Bridget from Guilty Gear was raised as a girl because of a superstition in his village about having same gender twins. His story mode actually has him setting off to become a bounty hunter to actually prove that he wouldn't be a curse on those he loved.
In the h-manga Boy Soprano, Akira is raised as a girl by his adoptive mother. Well, sort of — she makes him wear nothing but girls' clothing but openly acknowledges the fact that he's a boy in private. She's just very controlling; all this was a lead-up to him being enrolled in the all girls' school of which she's the principal, so she could keep a constant eye on him.
Marion Penrose from Unhinged.
An obscure manga called Princess Prince had Lawrence raised as a girl named Lori, due to a prophecy that if one of the twin sons born to the king was raised as a 'child of the moon' (read: a girl), there would be prosperity in the kingdom. (It doesn't help that of the two, he resembles his dead mother more.) Hilarity Ensues as Lawrence has a crush on a girl, is constantly hit on by men, and wonders what will happen when he hits puberty.
In the manhwa Angel Diary, Ee-Jung is raised as a girl because he is the Red Phoenix.
In Finder, Lynne was raised as a girl because her mother was from the Llaverac clan, where everyone, regardless of functional gender, has breasts and dresses and acts female. Lynne grew up to be an odd mixture.
In real life, David Reimer had his penis destroyed as a baby during a circumcision. John Money (a psychologist) then encouraged and oversaw David being sexually re-assigned to female and raised as such. John Money reported the case as a success, and used it as evidence that gender identity is environmental. However, it was later revealed that David had never identified as female, and began living as a male at age 15. He suffered from suicidal depression starting at 13, and eventually killed himself at 38.
Izumo Kunisaki in Kunisaki Izumo no Jijou
In the Brazilian soap-opera "Chocolate com Pimenta" (Chocolate with Pepper), there was a woman who suffered from a serious illness and promised to St. Bernardette she'd adopt a girl if she's cured. A maid working for the woman had a son and desired for him a better life so she passed the boy as a girl with no family and claimed the "girl's" name to be Bernardette. Not caring about her foster child more than required to keep good on her promise, she spent years unknowingly raising a boy until the truth came out and Bernardette renamed himself "Bernardo". The woman then started to think everything bad that happened to her was divine punishment for not keeping her promise.
The True Life episode "I'm Living As Someone I'm Not" had a woman who was biologically a man. She was raised by her mother as a girl after her mother saw that she was transgender around the age of 13. This counts as this trope because she did not really see herself as transgender, just as a woman and in fact had lived as a woman so long that she had even perfected techniques so that it looked like she had breasts, and it was impossible to tell that she was biologically a man (she just looked like a skinny tall woman). The story follows her trying to come out of the closet, so to speak.
Calliope/Cal, the central character of Middlesex, although being genetically male, is born with a condition that makes him look externally female. As neither the parents nor family doctor realised it until the child hit puberty, Callie was raised as a girl. Eventually, upon discovering his true nature he comes to terms with his male identity. Note that although Cal's parents specifically intended to have a baby girl (their first child was a boy), they obviously didn't intend to force her to fake her gender identity.
The Nostalgia Critic: "Granted, I didn't grow up as a girl. At least, not for long. *Beat* I-I have a history."
Canadian superhero Green Gables from the Spinnerette webcomic may possibly be an example. While he has not made mention of his childhood, he stated that the Green Gables persona is a matriarcal legacy in his family, leaving his origin hanging somewhere between this trope and Wholesome Crossdresser.
In Perfect Peace by Daniel Black, the mother of a black Southern family raises her youngest son as a girl. She manages to get away with this for 7 years because she makes sure to be the only one who changes her clothes, helps her in the bathroom etc.

Mixed Examples

Anime and Manga

Urusei Yatsura has Ryuunosuke, a girl raised as a boy, and Nagisa, a boy raised as a girl.
Hansel and Gretel in Black Lagoon were raised to role-play whatever sick scenarios their owner created for them and as the result learned to "switch" gender at will, becoming a dual Creepy Crossdresser.
Umineko no Naku Koro ni has Shannon/Yasu/Kanon, though which way this goes depends on how the reader interprets which personality is dominant and what his/her original gender was. Note that a large reason that no one can tell due to secondary attributes is the fact that they were injured at a young age in a way that is implied to have made them infertile and would naturally mess with the way that they developed.
Axis Powers Hetalia has Hungary growing up living as a boy (although it's also mentioned that she thought she really was one and thought that "we all get one when we get older anyway"; fandom opinions on Hungary's gender range from Tomboy to Transsexualism) and North Italy being raised as a girl by Austria and the aforementioned Hungary.

Literature

The Japanese classic Torikaebaya Monogatari has a brother and sister raised in this way. They go on to careers at court in those roles, but eventually switch to match their birth genders.
- The story has been adapted as a manga under the title of The Change!.

Real Life

This happens to almost every transgender person ever. Usually not due to malice, just because their parents don't know yet.