  Inspired by the neutrino oscillations data, we consider the exact $\mu-\tau$
symmetry, implemented at the level of the neutrino mass matrix, as a good
initial framework around which to study and describe neutrino phenomenology.
Working in the diagonal basis for the charged leptons, we deviate from
$\mu-\tau$ symmetry by just modifying the phases of the neutrino mass matrix
elements. This deviation is enough to allow for a non-vanishing neutrino mixing
entry $|V_{e3}|$ (i.e. $\theta_{13}$) but it also gives a very stringent (and
eventually falsifiable) prediction for the atmospheric neutrino mixing element
$|V_{\mu3}|$ as a function of $|V_{e3}|$. The breaking by phases is
characterized by a single phase and is shown to lead to interesting lower
bounds on the allowed mass of the lightest neutrino depending on the ordering
of neutrino masses (normal or inverted) and on the value of the Dirac ${\cal
CP}$ violating phase $\delta_{CP}$. The allowed parameter space for the
effective Majorana neutrino mass $m_{ee}$ is also shown to be non-trivially
constrained.
