  In an attempt to uncover any underlying structure in the neutrino mass
matrix, we discuss the possibility that the ratios of elements of its Majorana
mass matrix are equal. We call this "strong scaling Ansatz" for neutrino masses
and study its phenomenological implications. Of three possible independent
scale invariant possibilities, only one is allowed by current data, predicting
in a novel way the vanishing of U_{e3} and an inverted hierarchy with the
lightest neutrino having zero mass. The Ansatz has the additional virtue that
it is not affected by renormalization running. We also discuss explicit models
in which the scaling Ansatz is realized.
