Classification of congruent integer matrices I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known invariants that completely characterize congruency via unimodular integer matrices?
More precisely: Are there invariants under congruency of $A$ and $B$ that completely characterize whether or not some unimodular matrix $P\in\text{GL}_{2n}(\mathbb{Z})$ exists such that $P\cdot A\cdot P^T = B ?$
We can assume $A$ and $B$ to be non-singular, if it makes an answer possible.
 A: My Ph.D. thesis studied this and related questions from the perspective of arithmetic invariant theory.  (There's also various other literature out there addressing related questions from a different number of directions; my contribution was mainly putting this material into the context of arithmetic invariant theory.)
One thing to note is that all skew-symmetric unimodular matrices are congruent, and so one can fix $A-A^T = B-B^T=J$, and change the problem to ${\rm Sp}_{2n}(\Bbb Z)$-congruence of matrices $P$ with $P - P^T = J$.  I observed in section 3.3 of the thesis that these congruence classes are in bijection with what I called "conjugate-self-balanced" (CSB) modules over the ring $\Bbb Z[t]/f(t)$ where $f(t) = det(t J - P)$ (this is the Alexander polynomial up to a change of variables).
(I came up with the terminology CSB modules based on analogy with prior work of Wood -- however, similar objects are also called "polarized modules" in the context of the classification of abelian varieties over finite fields.  There's recently been a lot of computational work in that direction, and one interesting question is whether these tools could be adapted to work with Seifert pairings.)
So this is formally an answer to your question, but to actually do anything with this, you need to understand CSB modules over $\Bbb Z[t]/f(t)$ -- which may not be a Dedekind domain, or even a domain, or may even contain nilpotents -- meaning that even completely classifying modules over this ring may be messy even before bringing in the extra CSB structure.
