Invertus

You control "2" characters who each start one opposite halves of the screen. Actions are mirrored not just left to right but up and down.

Imagine: Character on left walks to the right and reaches a hill, and his compatriot on the right walks left and in the same spot on his half of the screen reaches a cliff. The leftward character is blocked and can't jump high enough. BUT the cliff in front of the righthand character is longer than the height of the hill. So player switches to righthand character, and proceeds to plummet to his death, BUT that got the left bound character to jump (reverse action of fall) high enough to get over the hill.

Now the right-character respawns (not sure if he spawns in the starting point or where he died).

Anyways, I'd imagine it's something like N++, Cursor*10, Portal, Braid, Time Donkey... One of those sorts of platforming games that add a twist.

Hell, maybe there's already a game like that (it wouldn't surprise me). But I like the idea and it could present some interesting level design challenges. Especially if you added powerups that could add features along the way.