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.
