Finding solutions to a symmetric divisibility condition $x\mid p(y),\;y\mid p(x)$

Question

In general, are there strategies for finding all integers $x$ and $y$ such that $x \mid p(y)$ and $y \mid p(x)$ for some polynomial $p$ with integer coefficients? For example, could we find all integers $x$ and $y$ (WLOG let $x \leq y$) such that $x \mid 2y^2-2y+1$ and $y \mid 2x^2-2x+1$. I know there are nontrivial solutions for this example: $(1, 1)$, $(17, 109)$, and $(29, 125)$ all satisfy these conditions, although I can't prove that there are or aren't any more solutions than that. Trying to express these conditions in equation form (i.e. $2y^2-2y+1 = kx$, $2x^2-2x+1 = my$) and eliminating variables doesn't seem to get me very far.