What is this symmetric-like property of $n$-ary relations?

Question

Let's consider that there is a set $S$ for which $R \subseteq S \times S \times S$ is a relation on that set. With $x,y,z \in S$, imagine that the following holds:

$$R(x, y, z) \iff R(x, z, y) \iff R(y, x, z) \iff R(y, z, x) \iff R(z, x, y) \iff R(z, y, x)$$

This 3-ary relation has a property that reminds me of symmetric relations in which the order of the elements doesn't matter. This property can hold non-trivially for arities $n>1$, and still hold for $n=0$ or $n=1$ in a trivial way, making it something rather general. What is this property called of an $n$-ary relation?

Answer 1

Well, there is the notion of symmetric function.

An $n$-ary function $f(x_1,\ldots,x_n)$ is symmetric if its value is the same no matter the order of its arguments.

Symmetric relations can be defined similarly - after all, functions are special relations.

Answer 2

The following is a published definition for a symmetric $n$-ary relation, with the qualifier of strongly to denote that all permutations must hold.

[An $n$-ary (finitary) relation $\rho$ is said to be] strongly symmetric if $(x_1, \cdots, x_n) \in \rho$ implies $(x_{\sigma(1)}, \cdots, x_{\sigma(n)}) \in \rho$ for any permutation $\sigma$ of the set $\{ 1, \cdots, n\}$.

Reference

https://www.sciencedirect.com/science/article/pii/S0195669809001589