  We consider the flavor symmetry L_mu - L_tau for the neutrino mass matrix.
The most general neutrino mass matrix conserving L_mu - L_tau predicts
quasi-degenerate neutrino masses with one maximal and two zero mixing angles.
The presence of L_mu - L_tau can also be motivated by the near-bimaximal form
of the neutrino mixing matrix. Furthermore, it is a special case of mu-tau
symmetric mass matrices. Breaking the flavor symmetry by adding a small
flavor-blind term to the neutrino mass matrix and/or by applying radiative
corrections is shown to reproduce the observed neutrino oscillation
phenomenology. Both the normal and inverted mass ordering can be accommodated
within this scheme. Moderate cancellation for neutrinoless double beta decay is
expected. The observables U_{e3}^2 and |1/2 - \sin^2\theta_{23}| are
proportional to the inverse of the fourth power of the common neutrino mass
scale. We comment on whether the atmospheric neutrino mixing is expected to lie
above or below pi/4. We finally present a model based on the see-saw mechanism
which generates a light neutrino mass matrix with an (approximate) L_mu - L_tau
flavor symmetry. This is a minimal model with just one standard Higgs doublet
and three heavy right-handed neutrinos. It needs only small values for the soft
L_mu - L_tau breaking terms to reproduce the phenomenological viable mass
textures analyzed.
