We discuss some Z_N^L x Z_N^R orbifold compactifications of the type IIBsuperstring to D= 4,6 dimensions and their type I descendants. Although theZ_N^L x Z_N^R generators act asymmetrically on the chiral string modes, theyresult into left-right symmetric models that admit sensible unorientablereductions. We carefully work out the phases that appear in the modulartransformations of the chiral amplitudes and identify the possibility ofintroducing discrete torsion. We propose a simplifying ansatz for theconstruction of the open-string descendants in which the transverse-channelKlein-bottle, annulus and Moebius-strip amplitudes are numerically identical inthe proper parametrization of the world-sheet. A simple variant of the ansatzfor the Z_2^L x Z_2^R orbifold gives rise to models with supersymmetry breakingin the open-string sector.