co.combinatorics - What can I say about the permutation $alphabeta$ if I know the permutation $betaalpha$?

I'm looking into a secret sharing scheme that has a secret permutation $theta$ which has the cycle structure (n/2)+(n/2) (i.e. two (n/2)-cycles).

The permutation $theta$ is decomposed into two permutations $alpha$ and $beta$, where $alpha$ is generated uniformly at random. So with knowledge of both $alpha$ and $beta$, we can find $theta$, while with knowledge of $alpha$ xor $beta$, we cannot find $theta$ (although, we could guess).

At this point, I want to make public $betaalpha(L)$ (L is actually a Latin square, but this is not too relevant for the question I want to ask). It is possible that an attacker could find $betaalpha$ from $betaalpha(L)$. However, I worry that knowledge of $betaalpha$ might give information about $theta$.

If I know $theta=alphabeta$, and I'm given the permutation $betaalpha$, what can I say about $theta$? (without a priori knowledge of $alpha$, $beta$ or $theta$)