  We study metric projections onto cones in the Wasserstein space of
probability measures, defined by stochastic orders. Dualities for backward and
forward projections are established under general conditions. Dual optimal
solutions and their characterizations require study on a case-by-case basis.
Particular attention is given to convex order and subharmonic order. While
backward and forward cones possess distinct geometric properties, strong
connections between backward and forward projections can be obtained in the
convex order case. Compared with convex order, the study of subharmonic order
is subtler. In all cases, Brenier-Strassen type polar factorization theorems
are proved, thus providing a full picture of the decomposition of optimal
couplings between probability measures given by deterministic contractions
(resp. expansions) and stochastic couplings. Our results extend to the forward
convex order case the decomposition obtained by Gozlan and Juillet, which
builds a connection with Caffarelli's contraction theorem. A further noteworthy
addition to the early results is the decomposition in the subharmonic order
case where the optimal mappings are characterized by volume distortion
properties. To our knowledge, this is the first time in this occasion such
results are available in the literature.
