The relation between the geometry of a two-dimensional sample and its equilibrium physical properties is exemplified here for a system of noninteracting electrons on a Mo¨bius strip. Dispersion relation for a clean sample is derived and its persistent current under moderate disorder is elucidated, using statistical analysis pertinent to a single sample experiment. The flux periodicity is found to be distinct from that in a cylindrical sample, and the essential role of disorder in the ability to experimentally identify a Mo¨bius strip is pointed out.