  Closed-form expressions are obtained for the generating function of
close-packed dimers on a $2M \times 2N$ simple quartic lattice embedded on a
M\"obius strip and a Klein bottle. Finite-size corrections are also analyzed
and compared with those under cylindrical and free boundary conditions.
Particularly, it is found that, for large lattices of the same size and with a
square symmetry, the number of dimer configurations on a M\"obius strip is
70.2% of that on a cylinder. We also establish two identities relating dimer
generating functions for M\"obius strips and cylinders.
