Type IIB orientifolds with discrete torsion

We consider compact four-dimensional ${\mathbf{Z}}_{\mathbf{N}}\times {\mathbf{Z}}_{\mathbf{M}}$ type IIB orientifolds, for certain values of $N$ and $M$. We allow the additional feature of discrete torsion and discuss the modification of the consistency conditions arising from tadpole cancellation. We point out the differences between the cases with and without discrete torsion.