  In this paper, we exhibit a new family of martingale couplings between two
one-dimensional probability measures $\mu$ and $\nu$ in the convex order. This
family is parametrised by two dimensional probability measures on the unit
square with respective marginal densities proportional to the positive and
negative parts of the difference between the quantile functions of $\mu$ and
$\nu$. It contains the inverse transform martingale coupling which is explicit
in terms of the associated cumulative distribution functions. The integral of
$\vert x-y\vert$ with respect to each of these couplings is smaller than twice
the $W_1$ distance between $\mu$ and $\nu$. When $\mu$ and $\nu$ are in the
decreasing (resp. increasing) convex order, the construction is generalised to
exhibit super (resp. sub) martingale couplings.
