  Consider a multiperiod optimal transport problem where distributions
$\mu_{0},\dots,\mu_{n}$ are prescribed and a transport corresponds to a scalar
martingale $X$ with marginals $X_{t}\sim\mu_{t}$. We introduce particular
couplings called left-monotone transports; they are characterized equivalently
by a no-crossing property of their support, as simultaneous optimizers for a
class of bivariate transport cost functions with a Spence--Mirrlees property,
and by an order-theoretic minimality property. Left-monotone transports are
unique if $\mu_{0}$ is atomless, but not in general. In the one-period case
$n=1$, these transports reduce to the Left-Curtain coupling of Beiglb\"ock and
Juillet. In the multiperiod case, the bivariate marginals for dates $(0,t)$ are
of Left-Curtain type, if and only if $\mu_{0},\dots,\mu_{n}$ have a specific
order property. The general analysis of the transport problem also gives rise
to a strong duality result and a description of its polar sets. Finally, we
study a variant where the intermediate marginals $\mu_{1},\dots,\mu_{n-1}$ are
not prescribed.
