There are in fact uncountably many non-monotonic pairing functions. One interesting family is obtained by applying some permutation $p : \mathbb{N} \rightarrow \mathbb{N}$ to the $x$ and $y$ inputs to the Cantor pairing function. We could, for example, swap even and odd inputs:

$$eggCarton(x, y) = cantorPair\big(x + cos(\pi x), y + cos(\pi y)\big)$$

When plotted, the manifold has a signature “egg carton” shape: Egg carton pairing function

The path traced by connecting each successive output value resembles a weave structure:

Egg carton pairing function

More complex permutations of the input result in more intricate patterns. Here we swap pairs of adjacent rows and columns:

Double egg carton pairing function

These examples have no practical utility, but they certainly yield beautiful visualizations.

The source code used to generate all figures in this post can be found here.