Abstract: We provide a classification of generalized CP symmetries preserved by the neutrino mass matrix according to the number of zero entries in the associated transformation matrix. We determine the corresponding constrained form of the lepton mixing matrix, characterized by correlations between the mixing angles and the CP violating phases. We compare with the corresponding restrictions from current neutrino oscillation global fits and show that, in some cases, the Dirac CP phase characterizing oscillations is highly constrained. Implications for current and upcoming long baseline neutrino oscillation experiments T2K, NO nu A, and DUNE, as well as neutrinoless double beta decay experiments are discussed. We also study the cosmological implications of such schemes for leptogenesis.