  After the successful determination of the reactor neutrino mixing angle
\theta_13 ~ 0.16 \neq 0, a new feature suggested by the current neutrino
oscillation data is a sizeable deviation of the atmospheric neutrino mixing
angle \theta_23 from \pi/4. Using the fact that the neutrino mixing matrix U =
U_e^\dagger U_\nu, where U_e and U_\nu result from the diagonalisation of the
charged lepton and neutrino mass matrices, and assuming that U_\nu has a i)
bimaximal (BM), ii) tri-bimaximal (TBM) form, or else iii) corresponds to the
conservation of the lepton charge L' = L_e - L_\mu - L_\tau (LC), we
investigate quantitatively what are the minimal forms of U_e, in terms of
angles and phases it contains, that can provide the requisite corrections to
U_\nu so that \theta_13, \theta_23 and the solar neutrino mixing angle
\theta_12 have values compatible with the current data. Two possible orderings
of the 12 and the 23 rotations in U_e, "standard" and "inverse", are
considered. The results we obtain depend strongly on the type of ordering. In
the case of "standard" ordering, in particular, the Dirac CP violation phase
\delta, present in U, is predicted to have a value in a narrow interval around
i) \delta ~ \pi in the BM (or LC) case, ii) \delta ~ 3\pi/2 or \pi/2 in the TBM
case, the CP conserving values \delta = 0, \pi, 2\pi being excluded in the TBM
case at more than 4\sigma. In the addendum we discuss the implications of the
latest 2013 data.
