Here is an example in which the Mobius strip is also physically visible. Namely, the old eight-track tape system, on which I listened to the Cars and Led Zeppelin as a teenager, has an endless tape with one twist, giving the basic tape-position configuration space the nature of a Mobius strip. (The configuration of the head position keeps track of an additional four positions across the tape.)
http://upload.wikimedia.org/wikipedia/commons/e/e3/8track_inside.JPG
