a.k.a. Figurenmatch, Distortion Demon Square
Patented by Manfred Fritsche 1983.
left: shiny stickers 2007, other sides are red and green, plastic, 3.8";
right: Mefferts keychain purchased 2007, plastic, 2.25" edges)

This puzzle is manipulated by turning along a cutting plane (the tips don't turn). It can be hard to visualize what is going on after the puzzle gets into some strange shapes. It is fairly easy to get it back to the pyramid shape; Meffert's Page, Jaap's Page, and McFarren's Page present solutions where this is the first step, but then the final steps employ transformations that temporarily go through non-pyramid shapes. Using a plain colored version of this puzzle, here are some photos of some of the shapes the puzzle can take.

Further Reading
Mefferts Page, from:
Jaap's Page, from:
McFarren's Page, from:
Wikipedia Page, from:
Fritsche DE Patent, from: - patent no. DE3245341