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Update: Consarn it! Looks like someone's more or less beat me to this one, and since I wrote this three months after the game was showcased at the Independent Games Festival 2008, I'm probably liable for some huge lawsuit. Ah well. Curse you Fez and your beauty!

Inspired by the Victorian-era fourth-dimensional primer by Abbot, this game would be basically your typical 2D platformer, except that the 2D world would actually be the surface of some 3D object. For example, picture 1-1 in Mario 1, but about halfway through the camera zooms out to reveal that the level is really a ribbon that twists, turns, and flips in 3D.

The ribbon might cut and reform based on various player actions. Think of that level in Lost Levels where you had to follow the correct path in order to continue, or else it would go "bonk" and make you do it again. That level would be a Möbius strip, looping you infinitely until you took that one path, at which point the strip would cut itself and join onto the next segment. Alternatively, it could be a simple loop, with level segments alternating per side.

A level with Original Mario Bros-style horizontal wrapping could be a vertical tube.

It's a variation on Super Paper Mario and Mario Galaxy, to some extent. And remember those episodes of Looney Tunes where Daffy Duck or whoever would be fighting the animator? Maybe a bit of that...