Orientifolds with discrete torsion

We show how discrete torsion can be implemented in D=4, N=1 type IIB orientifolds. Some consistency conditions are found from the closed string and open string spectrum and from tadpole cancellation. Only real values of the 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(Gamma,R^)=H^2(Gamma,Z_2). Four different types of orientifold constructions are possible. We classify these models and give the spectrum and the tadpole cancellation conditions explicitly.