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We provide a classification of generalized CP symmetries preserved by theneutrino mass matrix according to the number of zero entries in the associatedtransformation matrix. We determine the corresponding constrained form of thelepton mixing matrix, characterized by correlations between the mixing anglesand the CP violating phases. We compare with the corresponding restrictionsfrom current neutrino oscillation global fits and show that, in some cases, theDirac CP phase characterizing oscillations is highly constrained. Implicationsfor current and upcoming long baseline neutrino oscillation experiments T2K,NO$\nu$A and DUNE, as well as neutrinoless double beta decay experiments arediscussed. We also study the cosmological implications of such schemes forleptogenesis.

Classifying CP transformations according to their texture zeros: theory and implications