  This paper proposes various nonparametric tools based on measure
transportation for directional data. We use optimal transports to define new
notions of distribution and quantile functions on the hypersphere, with
meaningful quantile contours and regions and closed-form formulas under the
classical assumption of rotational symmetry. The empirical versions of our
distribution functions enjoy the expected Glivenko-Cantelli property of
traditional distribution functions. They provide fully distribution-free
concepts of ranks and signs and define data-driven systems of (curvilinear)
parallels and (hyper)meridians. Based on this, we also construct a universally
consistent test of uniformity and a class of fully distribution-free and
universally consistent tests for directional MANOVA which, in simulations,
outperform all their existing competitors. A real-data example involving the
analysis of sunspots concludes the paper.
