  We consider the scenario in which neutrino data are explained by the
interplay of type I and II seesaw terms in the Majorana neutrino mass matrix
${\cal M}_\nu = {\cal M}_L - {\cal M}_D {\cal M}_R^{-1} {\cal M}_D^T$. We
construct a predictive model with ${\cal M}_L$ proportional to the unit matrix,
3 diagonal texture zeros in ${\cal M}_R$, and ${\cal M}_D$ diagonal. We show
how this pattern can be maintained by the non-Abelian discrete symmetry $A_4$,
and discuss its phenomenological consequences. It turns out that the two types
of seesaw give contributions of the same order to ${\cal M}_\nu$. In the CP
conserving case, we find $\sin\theta_{13}\approx 2/(\tan 2\theta_{23} \tan
2\theta_{12})$ and we predict inverted (normal) ordering of the mass spectrum
for $\tan^2\theta_{12} < 0.5 ~(>0.5)$.
