Is the function $f(x,y)$ describing the given graph is symmetric for any $x\neq y$? Or, only for $x=y$?

Question

We have an infinite periodic graph with the two lengths $x$ and $y$

and we have a function $f(x,y)$ that describes a specific property of this periodic structure.

Then, according to the symmetry of the graph, should the function $f(x,y)$ be symmetric under the permutation $\;x\leftrightarrow y\;$ for any $x\neq y$? Or, only for $x=y$?

Answer 1

Assuming the "periodic structure" has no notion of a "starting point" (or at least, if $f$ is invariant under horizontal translations of the structure), then $f$ should be invariant under swapping the lengths, since this just amounts to translating the structure by length $x$.