In addition to FlameBoy's writeup, you may also want to try to cut 1/3 from an edge, and continue cutting 1/3 from that edge until your scissors reach the point where you started. This also produces a very interesting result. Try various things, like cutting the result again. You get all sorts of unexpected suprises, or maybe they are expected if you are a topologist.

M.C. Escher drew (etched?) an interesting lattice moebius strip, with ants crawling on it. While one ant may look as if it is on an opposite side compared to another ant, they are actually all on the same side. This drawing is animated in the xscreensaver GL screensaver collection, where you can watch the ants walk around to a place on the "other side" of their starting position.