Lying in the Dirt Together

Okay, so our protagonist is minding his/her own business, usually walking around the town of his/her birth and/or recalling memories of a fonder time. Then, he/she sees the Worthy Opponent / The Rival / the Anti-Hero /etc. doing the same. Naturally, upon seeing each other, they decide to duke it out, more for the sake of recalling old times or bonding than actual fighting over good and evil. After exhausting themselves (because they're so evenly matched that neither can defeat the other), the two rest on the ground and just talk and bond, temporarily forgetting that they're supposed to be enemies or on different sides.

This is related to Go-Karting with Bowser; however, it is an exact moment that occurs several times in a similar fashion in different shows/videogames/novels, rather than the act of bonding with an enemy. Usually occurs with Vitriolic Best Buds.

Related to Protagonist-Centered Morality. If the bad guy shares a moment with the protagonist, it doesn't matter that he's done awful things to people off screen.

Examples of Lying in the Dirt Together include:

Anime & Manga

Naruto and Sasuke do this in Naruto; however, the two are on top of separate trees rather than laying on the ground.
Arihiko recollects a time when he did this with Shiki in his Day in The Limelight story in Tsukihime Kagetsu-Tohya.
Tomoya and Sunohara sort of do this in Clannad, although they do make up the morning afterward, laughing at their own bruised faces.
Kazuma and his father during the bath house episode of Kaze no Stigma.
The Negi/Rakan fight in Mahou Sensei Negima ends with one of these. Negi and Kotaro also occasionally do this.
Samurai Champloo: Jin and Mugen's final fight.
Eas and Cure Peach after their last fight in Fresh Pretty Cure.
England and Germany after playing soccer in Axis Powers Hetalia.

Film

Crouching Tiger, Hidden Dragon: When Jen and Lo fight in the desert, Jen eventually knocks him out only to collapse next to him in the sand.
Happens between the Narrator and Tyler Durden in Fight Club. Of course, later events utterly change the meaning of the scene.

Live Action TV

In the 1980s TV series V, Donovan didn't like Tyler, and Tyler didn't like Donovan. Things eventually come to a head in the last episode of the series where the two of them finally get into a fistfight. They manage to beat the snot out of each other with the result that both wind up lying on the ground next to one another. Somehow this generates respect for each other in the process.
Star Trek: The Original Series episode "Shore Leave". While exploring an alien planet Captain Kirk meets Finnegan, an upperclassman who tormented him at Starfleet Academy. Finnegan taunts Kirk into a prolonged fight, which ends with both of them too tired to continue. It's later revealed that Finnegan was a robot created based on Kirk's memories.
In Star Trek: The Next Generation episode "Family", Picard comes home to his fuming older brother who was always a jealous bully to him. They throw punches, eventually stopping to laugh at themselves while they lay in the mud.
Happens in the Quantum Leap episode "A Single Drop of Rain" between Sam and the brother of the man Sam leapt into.

Video Games

In Persona 4, Yosuke and the protagonist fight in the final scene of Yosuke's social link at his request.
Yuri and Flynn do this after fighting in Tales of Vesperia.
Never shown, but Marcus the super-mutant and Jacob the Brotherhood of Steel Paladin met in battle shortly after the Master's defeat at the end of Fallout 1. They ended up working together to found Broken Hills, discussing the philosophies of their respective sides.

Marcus: We tussled for a while... (chuckles) probably a day or two. Then we just started laughing: what was the point?

Literature

This occurred in the backstory of the David Drake - Eric Flint Belisarius Series—Raghunath Rao and Rana Sanga fought a famous single combat, with their respective armies watching, that lasted for hours, both of them losing their weapons and wrestling in the course of it. As an eyewitness reported:

"By the end of the day, both men were too weak and exhausted to lift an arm, or even stand. So they laid down side by side and continued their combat with words.... Sanga ... swears he would rather fight a tiger with his own teeth than face Rao again on the field of philosophy."

The Ur-Example is Gilgamesh and Enkidu.