  $\mu$-$\tau$ symmetry imposed on the neutrino mass matrix in the flavour
basis is known to be quite predictive. We integrate this very specific neutrino
symmetry into a more general framework based on the supersymmetric SO(10) grand
unified theory. As in several other models, the fermion mass spectrum is
determined by Hermitian mass matrices resulting from the renormalizable Yukawa
couplings of the 16-plet of fermions with the Higgs fields transforming as $10,
\bar{126},120$ representations of the SO(10) group. The $\mu$-$\tau$ symmetry
is spontaneously broken through the 120-plet. Consequences of this scheme are
considered for fermion masses using both type-I and type-II seesaw mechanism.
This scenario is shown to lead to a generalized CP invariance of the mass
matrices and vanishing CP violating phases if the Yukawa couplings are
invariant under the $\mu$-$\tau$ symmetry. Small explicit breaking of the
$\mu$-$\tau$ symmetry is then shown to provide a very good understanding of all
the fermion masses and mixing. Detailed fits to the fermion spectrum are
presented in several scenarios. One obtains a very good fit to all observables
in the context of the type-I seesaw mechanism but type-II seesaw model also
provides a good description except for the overall scale of the neutrino
masses. Three major predictions on the leptonic mixing parameters in the type-I
seesaw case are (1) the atmospheric mixing angle $\theta_{23}^{l}$ close to
maximal, (2) $\theta_{13}^{l}$ close to the present upper bound and (3)
negative but very small Dirac CP violating phase in the neutrino oscillations.
