  Neutrino oscillations data indicates that neutrino mixings are consistent
with an apparent $\nu_\mu - \nu_\tau$ exchange symmetry in neutrino mass
matrix. We observe that in the mininimally extended standard model with the
see-saw mechanism, one can impose $\mu\leftrightarrow\tau$ symmetry at the tree
level on all Lagrangian terms, but for the mass difference among $\mu$ and
$\tau$ leptons. In the absence of any new extra physics, this mass difference
becomes the only source for the breaking of such a symmetry, which induces, via
radiative corrections, small but predictable values for $\theta_{13}$, and for
the deviation of $\theta_{ATM}$ from maximallity. In the CP conserving case,
the predictions only depend on neutrino mass hierarchy and may provide a unique
way to test for new physics with neutrino experiments.
