  We find a realization of the $Z_2$-symmetry in the neutrino mass matrix which
expresses a rotation of the $\mu-\tau$ symmetry and is able to impose a generic
smallest mixing angle, in contrast to a zero-value predicted by the usual
non-rotated form of the $\mu-\tau$ symmetry. We extend this symmetry for the
lepton sector within type-I seesaw scenario, and show it can accommodate the
mixing angles, the mass hierarchies and the lepton asymmetry in the universe.
We then study the effects of perturbing the specific form of the neutrino mass
matrix imposed by the symmetry and compute the resulting mixing and mass
spectrum. We trace back this "low-scale" perturbation to a "high-scale"
perturbation, and find realizations of this latter one arising from exact
symmetries with an enriched matter content.
