  The RV coefficient measures the similarity between two multivariate
configurations, and its significance testing has attracted various proposals in
the last decades. We present a new approach, the invariant orthogonal
integration, permitting to obtain the exact first four moments of the RV
coefficient under the null hypothesis. It consists in averaging along the Haar
measure the respective orientations of the two configurations, and can be
applied to any multivariate setting endowed with Euclidean distances between
the observations. Our proposal also covers the weighted setting of observations
of unequal importance, where the exchangeability assumption, justifying the
usual permutation tests, breaks down.
  The proposed RV moments express as simple functions of the kernel eigenvalues
occurring in the weighted multidimensional scaling of the two configurations.
The expressions for the third and fourth moments seem original. The first three
moments can be obtained by elementary means, but computing the fourth moment
requires a more sophisticated apparatus, the Weingarten calculus for orthogonal
groups. The central role of standard kernels and their spectral moments is
emphasized.
