The PMNS neutrino mixing matrix UPMNS is in general a product of two unitary matrices Ulep and Uν arising from the diagonalization of the charged lepton and neutrino mass matrices, U PMNS=U †lepU ν. Assuming that Uν is a bimaximal mixing matrix, we investigate the possible forms of Ulep. We identify three possible generic structures of Ulep, which are compatible with the existing data on neutrino mixing. One corresponds to a hierarchical "CKM-like" matrix. In this case relatively large values of the solar neutrino mixing angle θsol, and of | Ue3 | 2≡|( UPMNS) e3 | 2, are typically predicted, tan 2θsol≳0.42, | Ue3 | 2≳0.02, while the atmospheric neutrino mixing angle θatm can deviate noticeably from π/4, sin 22 θatm≳0.95. The second corresponds to one of the mixing angles in Ulep being equal to π/2, and predicts practically maximal atmospheric neutrino mixing sin 22 θatm≃1. Large atmospheric neutrino mixing, sin 22 θatm≳0.95, is naturally predicted by the third possible generic structure of Ulep, which corresponds to all three mixing angles in Ulep being large. We focus especially on the case of CP-non-conservation, analyzing it in detail. We show how the CP-violating phases, arising from the diagonalization of the neutrino and charged lepton mass matrices, contribute to the measured neutrino mixing observables.