Let $(X_1,Y_1),\dots (X_n, Y_n)$ be an i.i.d. sequence of vectors. Note that $X_i$ need not be independent of $Y_i$ in general. We consider $X_i$ and $Y_i$ continuous so their rankings are almost-surely uniquely well-defined.
Let $r_X(i) \in \{ 1,\dots, n\}$ be the ranking 'from the bottom' of $X_i$ among $X_1,\dots, X_n$; so denoting $X_{(j)}$ as the $j$-th order statistic (with $X_{(1)} < X_{(2)} < \dots < X_{(n)}$), we have $r_X(X_{(j)}) = j$.
Similarly, let $r_Y(i)$ denote the ranking from the botoom of $Y_i$ among $Y_1,\dots, Y_n$.
Thus, we have that $(r_X(1),\dots, r_X(n))$ and $(r_Y(1), \dots, r_Y(n))$ are random permutations of $(1,\dots,n)$. In fact, since the $(X_i,Y_i)$ are i.i.d., we know that the marginal distributions of $r_X$ and $r_Y$ (seen as random permutations) are uniform over the symmetric group $S_n$.
I am wondering if we can say anything about the joint distribution of $(r_X, r_Y)$, or if there is any literature on this topic. I am not sure where to start with this, other than in the trivial cases.
For concreteness, one might want to consider the case where $Y_i = X_i + \varepsilon_i$, where $\varepsilon_i$ are small perturbations independent of the $X_i$.