  Extensions of the standard model with a $U(1)$ gauge symmetry contain gauge
invariant kinetic mixing, $\sin\chi$, and gauge non-invariant mass mixing,
$\delta M^2$, between the hypercharge and the new gauge boson $Z'$. These
represent a priori incalculable but phenomenologically important parameters of
the theory. They become calculable if there exist spontaneously or softly
broken symmetries which forbid them at tree level but allow their generation at
the loop level. We discuss various symmetries falling in this category in the
context of the gauged $L_\mu-L_\tau$ models and their interplay with lepton
mixing. It is shown that one gets phenomenologically inconsistent lepton mixing
parameters if these symmetries are exact. Spontaneous breaking of these
symmetries can lead to consistent lepton mixing and also generates finite and
calculable values of these parameters at one or two loop order depending on the
underlying symmetry. We calculate these parameters in two specific cases: (i)
the standard seesaw model with $\mu$-$\tau$ symmetry broken by the masses of
the right-handed neutrinos and (ii) in a model containing a pair of vectorlike
charged leptons which break $\mu$-$\tau$ symmetry. In case (i), the
right-handed neutrinos are the only source of gauge mixing. The kinetic mixing
parameters are suppressed and vanish if the right-handed neutrinos decouple
from the theory. In contrast, there exists a finite gauge mixing in case (ii)
which survives even when the masses of vectorlike leptons are taken to
infinity, exhibiting non-decoupling behaviour. The seesaw model discussed here
represents a complete framework with practically no kinetic mixing and hence
can survive a large number of experimental probes used to rule out specific
ranges in the coupling $g^\prime$ and mass $M_{Z^\prime}$. The model can
generate non-universality in tau decays, which can be tested in future
experiments.
