  We are interested in martingale rearrangement couplings. As introduced by
Wiesel [37] in order to prove the stability of Martingale Optimal Transport
problems, these are projections in adapted Wasserstein distance of couplings
between two probability measures on the real line in the convex order onto the
set of martingale couplings between these two marginals. In reason of the lack
of relative compactness of the set of couplings with given marginals for the
adapted Wasserstein topology, the existence of such a projection is not clear
at all. Under a barycentre dispersion assumption on the original coupling which
is in particular satisfied by the Hoeffding-Fr\'echet or comonotone coupling,
Wiesel gives a clear algorithmic construction of a martingale rearrangement
when the marginals are finitely supported and then gets rid of the finite
support assumption by relying on a rather messy limiting procedure to overcome
the lack of relative compactness. Here, we give a direct general construction
of a martingale rearrangement coupling under the barycentre dispersion
assumption. This martingale rearrangement is obtained from the original
coupling by an approach similar to the construction we gave in [24] of the
inverse transform martingale coupling, a member of a family of martingale
couplings close to the Hoeffding-Fr\'echet coupling, but for a slightly
different injection in the set of extended couplings introduced by Beiglb\"ock
and Juillet [9] and which involve the uniform distribution on [0, 1] in
addition to the two marginals. We last discuss the stability in adapted
Wassertein distance of the inverse transform martingale coupling with respect
to the marginal distributions.
