Dual string pairs with N = 1 and N = 2 supersymmetry in four dimensions

Based on a simple adiabatic argument and by considering the heterotic string counterpart of certain symmetries of Type II superstrings such as (-1)FL and orientation reversal, we construct orbifold candidates for dual pairs of heterotic and Type II string theories with N = 2 and N = 1 supersymmetry. We also analyze from a similar point of view the K3 fibrations that enter in recently proposed N = 2 candidates and use this structure together with certain orientation-reversing symmetries to construct N = 1 dual pairs. These pairs involve generalizations of Type I vacua which can be equivalent to E 8 × E 8 heterotic strings, while standard Type I vacua are related to SO(32).