How do I make a 2d side-scrolling level fold in 3d ways?

If you've read A Wrinkle in Time, you should be familiar with the concept. Imagine the level is drawn on a sheet of paper. You hit a switch, and the paper folds into a box. If the player walks into an edge of the box,he appears on the other face that the edge is touching. I am having problems with progamming this concept. I attempted to split the level into triangular modules that rearrange themselves, but it got way too complicated. Is there a simpler way, such as making a texture that acts as a window to a 2d game, then translate the player's point on the plane where faces are glued together?