  The need to test whether two random vectors are independent has spawned a
large number of competing measures of dependence. We are interested in
nonparametric measures that are invariant under strictly increasing
transformations, such as Kendall's tau, Hoeffding's D, and the more recently
discovered Bergsma--Dassios sign covariance. Each of these measures exhibits
symmetries that are not readily apparent from their definitions. Making these
symmetries explicit, we define a new class of multivariate nonparametric
measures of dependence that we refer to as Symmetric Rank Covariances. This new
class generalises all of the above measures and leads naturally to multivariate
extensions of the Bergsma--Dassios sign covariance. Symmetric Rank Covariances
may be estimated unbiasedly using U-statistics for which we prove results on
computational efficiency and large-sample behavior. The algorithms we develop
for their computation include, to the best of our knowledge, the first
efficient algorithms for the well-known Hoeffding's D statistic in the
multivariate setting.
