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

Backward and Forward Wasserstein Projections in Stochastic Order