  It is known that spheres have negative type, but only subsets with at most
one pair of antipodal points have strict negative type. These are conditions on
the (angular) distances within any finite subset of points. We show that
subsets with at most one pair of antipodal points have strong negative type, a
condition on every probability distribution of points. This implies that the
function of expected distances to points determines uniquely the probability
measure on such a set. It also implies that the distance covariance test for
stochastic independence, introduced by Sz\'ekely, Rizzo and Bakirov, is
consistent against all alternatives in such sets. Similarly, it allows tests of
goodness of fit, equality of distributions, and hierarchical clustering with
angular distances. We prove this by showing an analogue of the Cram\'er--Wold
theorem.
