Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal B}$.

Assume the following:

1. $f(X)$ is independent of $Y$
2. $g(Y)$ is independent of $X$

Can we say that there exist independent variables $A, B, C$ and functions $h_1, h_2$ such that the joint distribution of f(X), g(Y), X, Y is same as $A$, $B$, $h_1(A, C)$ and $h_2(B, C)$?

The intuition behind the claim is that conditioned on $f(X)$ and $g(Y)$, the joint distribution of $X, Y$ should be such that $X$ depends only on $f(X)$, and $Y$ depends only on $g(Y)$. The marginal distribution of one of $X$ or $Y$ conditioned on $f(X)$ and $g(Y)$ of course satisfies this property by the given independence assumptions. The question is whether we can show such a statement for the joint distribution..

If not, then is there a counter-example? Of course, coming up with a counter-example might be hard since one needs to show that it is not true for any choice $A, B, C$, $h_1$ and $h_2$, but maybe there is some intuitive reasoning why this is not true..