Form Invariance of the Neutrino Mass Matrix

Consider the most general $3 \times 3$ Majorana neutrino mass matrix $\cal M$. Motivated by present neutrino-oscillation data, much theoretical effort is directed at reducing it to a specific texture in terms of a small number of parameters. This procedure is often {\it ad hoc}. I propose instead that for any $\cal M$ one may choose, it should satisfy the condition $U {\cal M} U^T = {\cal M}$, where $U \neq 1$ is a specific unitary matrix such that $U^N$ represents a well-defined discrete symmetry in the $\nu_{e,\mu,\tau}$ basis, $N$ being a particular integer not necessarily equal to one. I illustrate this idea with a number of examples, including the realistic case of an inverted hierarchy of neutrino masses.