[math-fun] Moebius band rotations puzzle

Moebius band rotations puzzle: ----- Find explicitly a subset of the space SO(3) of rotations of 3-space that is topologically a (closed) Moebius band. ----- Bonus question: ----- Find explicitly three subsets M_1, M_2, M_3 of SO(3) that are each topologically a closed Moebius band, and such that the intersection of any two of them is topologically a disk. Do this so the solution is minimal, in the sense that replacing any M_j by a proper subset M_j' will not satisfy the conditions. ----- —Dan