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 $\mathcal{M}$ one may choose, it should satisfy the condition $U {\cal M} U^T = {\cal M}$,where$U \neq 1$isaspecificunitarymatrixsuchthat$U^N$representsawell-defineddiscretesymmetryinthe$\nu_{e,\mu,\tau}$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.