Most common pairing functions $\pi: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, like the Cantor pairing function, are not symmetric (i.e $\pi(k_1, k_2) \neq \pi(k_2, k_1)$), it seems to me. Xie 2021 defines a symmetric one, but "just" on the positive integers.
Question: are there any known symmetric pairing functions defined on the natural numbers?