A canonical representative, for this equivalence relation on matrices
Here are some ideas to get started. First, we can arrange that the diagonal is of the form $0 \cdots 0 1 \cdots 1$. This breaks the original problem into two new problems which are equivalent to each other. Let's concentrate on the part with zeroes on the diagonal. We can sort the rows according to Hamming weight, and can arrange the first row to be of the form $0\cdots 0 1\cdots 1$, say the one with minimal weight. If it is the only row having the same weight, then again we have broken the problem into two parts, and can continue recursively. If there are several rows with the same minimal weight, we can try tie-breaking using column weights, but this is not always enough. That's where this approach gets stuck, though perhaps one could be more creative and extract a normal form from these ideas.