  In the limit of an approximate $\mu-\tau$ symmetry in the neutrino mass
matrix, we explore deviations to the Tri-Bi-Maximal mixing pattern in the
neutrino sector. We consider two different ansatzes for the corrected pattern
to predict the current values of neutrino mixing parameters. We show that it is
possible to constrain the Majorana $CP$ phases by studying their correlation to
the mixing parameters and we study their effects on neutrinoless double beta
decay observables. These predictions are sharp for the quasi-degenerate
ordering and can be tested in upcoming experiments.
