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In this paper we study the problem of measuring and testing joint independence for a collection of multivariate random variables. Using the emerging theory of optimal transport (OT) based multivariate ranks, we propose a distribution-free test for multivariate joint independence. Towards this we introduce the notion of rank joint distance covariance (RJdCov), the higher-order rank analogue of the celebrated distance covariance measure, that captures the dependencies among all the subsets of the variables. The RJdCov can be easily estimated from the data without any moment assumptions and the associated test for joint independence is universally consistent. We can calibrate the test without any knowledge of the (unknown) marginal distributions (due to the distribution-free property), both asymptotically and in finite samples. In addition to being distribution-free and universally consistent, the proposed test is also statistically efficient, that is, it has non-trivial asymptotic (Pitman) efficiency. We demonstrate this by computing the limiting local power of the test for both mixture alternatives and joint Konijn alternatives. We also use the RJdCov measure to develop a method for independent component analysis (ICA) that is easy to implement and robust to outliers and contamination. Extensive simulations are performed to illustrate the efficacy of the proposed test in comparison to other existing methods. Finally, we apply the proposed test to learn the higher-order dependence structure among different US industries based on stock prices.