A generalization of the concept of Wigner's symmetrical operator, i.e., of the operator invariant under symmetry transformations forming a group, namely , is proposed here. Symmetrical operators in the present general sense do constitute a set whose elements are mutually commuting and equivalent to each other under transformations . A generalization of a well known theorem attributed to Wigner is established: for a given operator Â and its equivalent ones Â[i] = Ŝ ÂŜi, symmetrical as previously defined, it is always possible to find a complete set of common eigenfunctions that form canonical basis sets for representations of induced by the largest subgroup whose elements keep Â invariant (the particular case reduces to Wigner's theorem itself). The formal generalization presented here opens the possibility of describing and exploiting two complementary aspects of symmetry: invariance and equivalence, within a unified theoretical approach. It is also shown that the symmetry properties of the stationary solutions of Boys' localization functional obey a straightforward extension of this theorem. A few examples are provided to highlight the implications of this alternative way to look at symmetry in quantum systems. © 2012 Wiley Periodicals, Inc.