  A classical result of Strassen asserts that given probabilities $\mu, \nu$ on
the real line which are in convex order, there exists a \emph{martingale
coupling} with these marginals, i.e.\ a random vector $(X_1,X_2)$ such that
$X_1\sim \mu, X_2\sim \nu$ and $E[X_2|X_1]=X_1$. Remarkably, it is a non
trivial problem to construct particular solutions to this problem. In this
article, we introduce a family of such martingale couplings, each of which
admits several characterizations in terms of optimality properties / geometry
of the support set / representation through a Skorokhod embedding. As a
particular element of this family we recover the (left-) curtain martingale
transport, which has recently been studied \cite{BeJu16, HeTo13, CaLaMa14,
BeHeTo15} and which can be viewed as a martingale analogue of the classical
monotone rearrangement. As another canonical element of this family we identify
a martingale coupling that resembles the usual \emph{product coupling} and
appears as an optimizer in the general transport problem recently introduced by
Gozlan et al. In addition, this coupling provides an explicit example of a
Lipschitz-kernel, shedding new light on Kellerer's proof of the existence of
Markov martingales with specified marginals.
