  The purpose of this paper is to study the equivalence relation on unitary
bases defined by R. F. Werner [{\it J. Phys. A: Math. Gen.} {\bf 34} (2001)
7081], relate it to local operations on maximally entangled vectors bases, find
an invariant for equivalence classes in terms of certain commuting systems,
and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations
are given in the context of latin squares and projective representations as
well. Applications to quantum tomography are indicated.
