Z_N x Z_M orientifolds with and without discrete torsion

We discuss compact four-dimensional Z_N x Z_M type IIB orientifolds. We take a systematic approach to classify the possible models and construct them explicitely. The supersymmetric orientifolds of this type have already been constructed some time ago. We find that there exist several consistent orientifolds for each of the discrete groups Z_2 x Z_2, Z_2 x Z_4, Z_4 x Z_4, Z_2 x Z_6, Z_2 x Z_6' and Z_6 x Z_6 if anti-D5-branes are introduced. Supersymmetry is broken by the open strings ending on antibranes. The rank of the gauge group is reduced by a factor two if the underlying orbifold space has discrete torsion.