  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.
