In a theoretical framework independent of models, where the neutrinos are considered as Majorana particles and the mass matrix of charged leptons is diagonal, we perform a systematic study of so called CP symmetries. In the case that one remnant CP transformation is preserved by the neutrino mass matrix, the number of free parameters into flavor mixing matrix is reduce from six to four. In particular we propose a generalized mu-tau reflection symmetry to constrain the lepton flavor mixing parameters. We obtain a new correlation between the atmospheric mixing angle and the “Dirac” CP violation phase. Only in a specific limit our proposed CP transformation reduces to standard mu–tau reflection. The “Majorana” phases are predicted to lie at their CP-conserving values with important implications for the neutrinoless double beta decay rates. We also study the phenomenological implications of our scheme for present and future neutrino oscillation experiments including T2K,NOvA and DUNE.