  Within the type-I seesaw framework with three heavy right chiral neutrinos
and in the basis where the latter and the charged leptons are mass diagonal, a
near `mu-tau' symmetry in the neutrino sector is strongly suggested by the
neutrino oscillation data. There is further evidence for a close to the
tribimaximal mixing pattern which subsumes `mu-tau' symmetry. On the other
hand, the assumption of a (maximally allowed) four zero texture in the Yukawa
coupling matrix Y_nu in the same basis leads to a highly constrained and
predictive theoretical scheme. We show that the requirement of an exact
`mu-tau' symmetry, coupled with observational constraints, reduces the `seventy
two' allowed textures in such a `Y_nu' to 'only four' corresponding to just two
different forms of the light neutrino mass matrix `m_nu'. The effect of each of
these on measurable quantities can be described, apart from an overall factor
of the neutrino mass scale, in terms of two real parameters and a phase angle
all of which are within very constrained ranges. The additional input of a
tribimaximal mixing reduces these three parameters to `only one' with a very
nearly fixed value. Implications for both flavored and unflavored leptogenesis
as well as radiative lepton flavor violating decays are discussed. We also
investigate the stability of these conclusions under small deviations due to
renormalization group running from a high scale where the four zero texture as
well as `mu-tau' symmetry or the tribimaximal mixing pattern are imposed.
