  Within the different patterns of the neutrino mixing matrix, the cobimaximal
mixing remains a plausible possibility for understanding the flavor structure
of neutrinos as it is consistent with current experimental data. Such a pattern
is related to a concrete form of the mass matrix, displaying a $\mu-\tau$
reflection symmetry, which has motivated many theoretical investigations in
recent years. In this paper, we discuss the effects of the Majorana phases over
a $\mu-\tau$ symmetric mass matrix obtained from a cobimaximal mixing matrix.
We investigate how these phases could be restricted through deviations from the
symmetric scenario and the most precise determinations of the mixing
parameters. Some of our relations could be tested with future results.
