Two characters who since they first met have been at odds with each other — or even active enemies — decide in the wake of some event or epiphany to wipe the slate clean and start their relationship anew. To symbolize this new beginning, they re-enact a new "first meeting" between themselves by shaking hands and trading introductions. Sometimes they also trade capsule descriptions of themselves.
Not to be confused with Relationship Reset Button.
Blast Hardcheese and Doctor Lady do this in Space Mutiny right after she shows off her, uh, dancing.
Invoked in Mr. and Mrs. Smith. Both John and Jane find out that the other is a paid assassin and, by extension, that everything they've told each other about themselves is a lie. Each starts telling the other the truth about their real selves, and John notes, "We're going to have to redo every conversation we've ever had."
In Shrek 2, Shrek and Fiona's dad get off on the wrong foot, since he wasn't expecting his daughter to marry an ogre, and it messes up the arrangment he had with the Fairy Godmother, who intended her son Prince Charming to rescue Fiona. As part of the Fairy Godmother's plan, King Harold apologizes to Shrek for his previous behavior, and asks if they can pretend the whole thing never happened. Since Shrek is feeling guilty for his own temper, he accepts.
Suggested in Wizard and Glass by Susan after she and the witch Rhea are at each other's throats about two minutes after they've met. Susan asks if they could "begin again"; Rhea says that's impossible but "perhaps we could go on better than we began." They don't.
One episode of Full House has a disastrous first meeting between the Tanners and Joey's new girlfriend due to her being nervous about living up to everything Joey had told the family about her. The episode ends with her re-introducing herself, and the Tanners saying, "Joey has told us absolutely nothing about you."
In The Most Happy Fella, Tony and Rosabella go through "Happy To Meet Your Acquaintance" in their first scene together since their wedding (which took place just after Tony was badly injured in a truck accident). Even more poignant is the final scene, where Tony recalls (in words and music) the note from him that Rosabella (or, to use her real name, Amy) read off a restaurant menu in the first scene.
Towards the end of the Sister arc in El Goonish Shive, Ellen and Nanase do this. This is an unusual example, because Ellen is a magical Opposite-Sex Clone of one of the main characters, and despite having all of Elliot's memories, Ellen is an entirely new and different person without a friend in the world, which is causing her quite a bit of angst. Slightly subverted, as Nanase doesn't get it at first. Ellen then goes on to reintroduce herself to the entire cast, with Nanase threatening to bitch slap anyone who laughs.
At the end of the fifth chapter of Cat Legend, Sarah does this with Dorian. The congeniality of their second introduction is especially ironic in light of the fact that they are both sporting highly visible facial injuries they inflicted on each other.