Closed-form expressions are obtained for the partition function of the Ising model on an MxN simple-quartic lattice embedded on a Möbius strip and a Klein bottle. The solutions all lead to the same bulk free energy, but for finite M and N the expressions are different depending on whether the strip width M is odd or even. Finite-size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions. Our results are consistent with the conformal field prediction of a central charge c=1/2, provided that the twisted Möbius boundary condition is regarded as a free or fixed boundary.