Given four distinct points $a,b,c,d$ on a circle, we can (speaking informally) choose in a unique way two pairs separating each other. For example the pairs $a,c$ and $b,d$ separate each other, if going around the circle starting from $a$, we pass the four points in the order $a\,b\,c\,d$ or (going the other way round) in the order $a\,d\,c\,b$. This is an informal description of a $4$-ary relation $S$ (in this case on the set of points of a circle), which is known in Geometry as the Separation relation.
Now $S$ is closely related to the group $D_8$ of symmetries of a square in the following sense: Given four elements $a_1,a_2,a_3,a_4$ (in this case points on our circle) and a permutation $\pi \in S_4$, we have $$S\,a_1a_2a_3a_4 \quad\Longrightarrow \quad ( S\,a_{\pi(1)}a_{\pi(2)} a_{\pi(3)}a_{\pi(4)} \, \Leftrightarrow \, \pi \in D_8) $$ (where $D_8$ is considered as subgroup of $S_4$).
My question: Has this sort of connection between a group $G$ and the symmetries of a relation $R$ been studied? Is there a term for this situation, e.g. “$R$ is $G$-symmetric“? I wouldn't want to make up a word if there is already one in usage.