In this talk we define a new class of multivariate nonparametric measures of dependence that we refer to as symmetric rank covariances. This new class generalizes many existing classical rank measures of dependence, such as Kendall's tau and Hoeffding's D, as well as the more recently discovered Bergsma--Dassios sign covariance. Symmetric rank covariances make explicit the implicit symmetries hidden in the standard definitions of the above measures and, in doing so, lead naturally to multivariate extensions of the Bergsma--Dassios sign covariance. Symmetric rank covariances may be estimated unbiasedly using U-statistics for which we provide results on computational efficiency, leveraging tools from computational geometry, 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.