  The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon,
1945) are among the first examples of a nonparametric test. These procedures --
based on signs, (absolute) ranks and signed-ranks -- yield distribution-free
tests for symmetry in one-dimension. In this paper we propose a novel and
unified framework for distribution-free testing of multivariate symmetry (that
includes central symmetry, sign symmetry, spherical symmetry, etc.) based on
the theory of optimal transport. Our approach leads to notions of
distribution-free generalized multivariate signs, ranks and signed-ranks. As a
consequence, we develop analogues of the sign and Wilcoxon signed-rank tests
that share many of the appealing properties of their one-dimensional
counterparts. In particular, the proposed tests are exactly distribution-free
in finite samples with an asymptotic normal limit, and adapt to various notions
of multivariate symmetry. We study the consistency of the proposed tests and
their behavior under local alternatives, and show that the proposed generalized
Wilcoxon signed-rank (GWSR) test is particularly powerful against location
shift alternatives. We show that in a large class of such models, our GWSR test
suffers from no loss in (asymptotic) efficiency, when compared to Hotelling's
$T^2$ test, despite being nonparametric and exactly distribution-free. An
appropriately score transformed version of the GWSR statistic leads to a
locally asymptotically optimal test. Further, our method can be readily used to
construct distribution-free confidence sets for the center of symmetry.
