The definition of symmetry proposed by you does not seem adequate.
Plummer's Cross is NOT symmetrical by that definition. Given
Plummer's Cross in any orientation, there are plenty of elements
of the 24-move whole-cube rotation group which will shift it to
something NOT isomorphic through a substitution of colors to the
original. Yet, we know, intuitively, that the CP is "highly
symmetric", and it is a local maximum. It is as though I were
saying that this equivalence of all twists "out" of this position
almost defined symmetry.....