I preface my question by stating that I have no scholarly expertise in mathematics. I have a Ph.D. in philosophy and have recently developed an interest in logic, but even logic is outside of my professional expertise. I can only hope that my ignorance in the matter that interests me is not beyond remedy.
As I understand, a relation is understood in discrete mathematics as a set of ordered n-tuples. So suppose that there are two ternary relations that consist of different triples on (is "on" the right preposition to use here?) the same set of objects. For example, suppose that, on the set of positive integers, R1 is the set of triples (x, y, z) such that x < y < z, while R2 is the set of triples (x, y, z) such that y < x < z. I would say that there is a one-to-one correspondence between members of R1 (is "members" the right word? I mean the triples that belong to R1) and members of R2 in accordance with a consistent rule of permutation. But I don't know if I am describing the relation between the two relations correctly. If I am not, I hope that my intended meaning is clear enough that someone can restate it for me in a correct fashion.
All of that is just to set up my main question, which is: Is there a simple way to describe the relation between R1 and R2? I have chosen a pair of ternary relations for my example, but the relations could be of any -arity. I deliberately did not use a binary relation, because that sort of case does not present the same difficulties. As I understand, two binary relations that stand in the sort of correspondence that I have described are said to be inverses of each other. (One would say "converses" in logic, but apparently "inverse" is the term standardly used in mathematics; please correct me if I am mistaken on this point.) Can one say that R1 and R2 are inverses of each other?
There sadly isn't a snappy term for this sort of relationship. One key point is that there are (unlike with a binary relation) too many ways to "rearrange the coordinates" - one for every permutation of $\{1,...,n\}$ - so we couldn't say that $R_2$ is "the" [something] of $R_1$.
I've seen the language "$R$ is a permutation of $S$" to mean that $R,S$ are relations of the same arity $n$ and there's some permutation $\pi$ of $\{1,...,n\}$ such that $$R(x_1,...,x_n)\iff S(x_{\pi(1)},...,x_{\pi(n)}),$$ but I don't think that's actually universal.
Incidentally, from a set-theoretic perspective a relation is identified with the set of tuples it holds on: so e.g. in the integers, "$<$" literally is the set $$\{\langle a,b\rangle\in\mathbb{Z}^2: a<b\}.$$ (Here "$\langle\cdot,\cdot\rangle$" is some fixed pairing function - the specific choice almost never matters.) So your language "members of $R_1$" is exactly correct.