  We obtain predictions for the Majorana phases $\alpha_{21}/2$ and
$\alpha_{31}/2$ of the $3\times 3$ unitary neutrino mixing matrix $U =
U_e^{\dagger} \, U_{\nu}$, $U_e$ and $U_{\nu}$ being the $3\times 3$ unitary
matrices resulting from the diagonalisation of the charged lepton and neutrino
Majorana mass matrices, respectively. We focus on forms of $U_e$ and $U_{\nu}$
permitting to express $\alpha_{21}/2$ and $\alpha_{31}/2$ in terms of the Dirac
phase $\delta$ and the three neutrino mixing angles of the standard
parametrisation of $U$, and the angles and the two Majorana-like phases
$\xi_{21}/2$ and $\xi_{31}/2$ present, in general, in $U_{\nu}$. The concrete
forms of $U_{\nu}$ considered are fixed by, or associated with, symmetries
(tri-bimaximal, bimaximal, etc.), so that the angles in $U_{\nu}$ are fixed.
For each of these forms and forms of $U_e$ that allow to reproduce the measured
values of the three neutrino mixing angles $\theta_{12}$, $\theta_{23}$ and
$\theta_{13}$, we derive predictions for phase differences $(\alpha_{21}/2 -
\xi_{21}/2)$, $(\alpha_{31}/2 - \xi_{31}/2)$, etc., which are completely
determined by the values of the mixing angles. We show that the requirement of
generalised CP invariance of the neutrino Majorana mass term implies $\xi_{21}
= 0$ or $\pi$ and $\xi_{31} = 0$ or $\pi$. For these values of $\xi_{21}$ and
$\xi_{31}$ and the best fit values of $\theta_{12}$, $\theta_{23}$ and
$\theta_{13}$, we present predictions for the effective Majorana mass in
neutrinoless double beta decay for both neutrino mass spectra with normal and
inverted ordering.
