  Using the seesaw mechanism and a discrete symmetry, we construct a class of
models for the neutrino mass matrix where the inverse of that matrix is the sum
of a mu-tau antisymmetric background and a perturbation. We consider various
possibilities for that perturbation. The simplest possible perturbations lead
to four-parameter neutrino mass matrices which are unable to fit the
experimental data. More complicated perturbations give rise to viable
six-parameter mass matrices; we present detailed predictions of each of them.
