Form Invariance of the Neutrino Mass Matrix

Consider the most general $3\times 3$ Majorana neutrino mass matrix $\mathcal{M}$. 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 ad hoc. I propose instead that for any $\mathcal{M}$ one may choose, it should satisfy the condition $U\mathcal{M}{U}^T=\mathcal{M}$, where $U\ne 1$ 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 1. I illustrate this idea with a number of examples, including the realistic case of an inverted hierarchy of neutrino masses.