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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\u2013Cantelli 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 article.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae026","type":"journal-article","created":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T04:59:20Z","timestamp":1714625960000},"page":"1172-1196","source":"Crossref","is-referenced-by-count":8,"title":["Nonparametric measure-transportation-based methods for directional data"],"prefix":"10.1093","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6599-7409","authenticated-orcid":false,"given":"M","family":"Hallin","sequence":"first","affiliation":[{"name":"ECARES and D\u00e9partement de Math\u00e9matique, Universit\u00e9 libre de Bruxelles , Brussels B-1050 ,","place":["Belgium"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Scitech Business, School of Management, University of Science and Technology of China , Hefei 230026, Anhui Province 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