Take all the $n\times n$ matrices of 0's and 1's and define an equivalence relation as follows: Two matrices are equal if there is a way to pass from the one to another by alternating the columns and the rows (acting by $S_n$ on the columns and on the rows).

Is there a good way to determine whether two such matrices are equal?

Are there any good invariants (polynomials, etc.)?

The obvious invariant is that the sum of the 1's on the rows and on the columns does not change.