Robust Comparable Pairs in Random Points

Let $\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_n$ be independent and uniformly distributed on $[0,1]^d$ (the $d$-dimensional unit hypercube). Two points $\mathbf{X}_i$ and $\mathbf{X}_j$ are called comparable if one dominates the other coordinatewise, i.e., either $\mathbf{X}_i \le \mathbf{X}_j$ in every coordinate or $\mathbf{X}_j \le \mathbf{X}_i$ in every coordinate. Find the expected number of comparable pairs.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.