Orientifolds with discrete torsion

We show how discrete torsion can be implemented in D = 4, N = 1 type-IIB or ientifolds. Some consistency conditions are found from the closed string an d open string spectrum and from tadpole cancellation. Only real values of t he discrete torsion parameter are allowed, i.e. epsilon = +/-1. Orientifold models are related to real projective representations. In a similar way as complex projective representations are classified by H-2 (Gamma, C*) = H-2 (Gamma, U(1)), real projective representations are characterized by H-2 (Ga mma, R*) = H-2(Gamma, Z(2)). Four different types of orientifold constructi ons are possible. We classify these models and give the spectrum and the ta dpole cancellation conditions explicitely.