Can you give examples for mechanical system that has a Mobius strip as their configuration space?
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.)
Imagine a rod of length d confined inside a spherical shell of diameter d and free to rotate within it. Provided the two ends of the rod are indistinguishable and the rod is cylindrically symmetric, the configuration space of the rod is the real projective plane (for an application of this sort of picture in nature, see this answer). If you glue a block to the inside surface of the sphere, this confines the rod further and now the configuration space is the Mobius strip.