Canonical permuation of a Symmetric matrix

In investigating How to generate the 8^th order symmetric binary matrices whose sum of absolute eigenvalues is 8? I wished to avoid considering matrices that differed only by the same permutation applied to their rows and columns.

My plan was to define a function perm transforming a matrix m to some canonical form and consider it only if perm(m)==m.

My naive approach defined

perm(u_?MatrixQ) := Module({ord}, ord = Ordering(u); #((ord)) & /@ u((ord)))

but I now realise that this doesn’t work as reordering the columns changes the orders of the rows. For example

m = {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 1, 
    1}, {0, 0, 1, 1, 0, 0}, {0, 0, 1, 0, 1, 1}, {0, 0, 1, 0, 1, 1}};

perm(m) == perm(perm(m))
(* False *)

Is there a simple way for defining and computing a canonical permutation?