  In view of the recent neutrino oscillation data pointing to a non-vanishing
value for the smallest mixing angle ($\theta_z$), we derive and find explicit
realizations of the $(Z_2)^3$ flavor symmetry which characterizes, for the
neutrino mass matrix, uniquely a variant of the tripartite form, originally
conceived to lead to the tri-bimaximal mixing with $\theta_z=0$, so that to
allow now for a non-tri-bimaximal pattern with non-zero $\theta_z$. We impose
this flavor symmetry in a setting including the charged leptons and we see that
it can make room, through higher order terms involving new SM-singlet scalars,
for the mass hierarchy of charged leptons. Moreover, within type-I seesaw
mechanism augmented with the flavor symmetry, certain patterns occurring in
both the Dirac and the Majorana neutrino mass matrices can accommodate all
types of mass hierarchies in the effective neutrino mass matrix, but no
lepton/baryon asymmetry can be generated. Finally, we discuss how type-II
seesaw mechanism, when supplemented with the flavor symmetry, could be used to
interpret the observed baryon asymmetry through leptogenesis.
