  It is well known that martingale transport plans between marginals
$\mu\neq\nu$ are never given by Monge maps -- with the understanding that the
map is over the first marginal $\mu$, or forward in time. Here, we change the
perspective, with surprising results. We show that any distributions $\mu,\nu$
in convex order with $\nu$ atomless admit a martingale coupling given by a
Monge map over the second marginal $\nu$. Namely, we construct a particular
coupling called the barcode transport. Much more generally, we prove that such
``backward Monge'' martingale transports are dense in the set of all martingale
couplings, paralleling the classical denseness result for Monge transports in
the Kantorovich formulation of optimal transport. Various properties and
applications are presented, including a refined version of Strassen's theorem
and a mimicking theorem where the marginals of a given martingale are
reproduced by a ``backward deterministic'' martingale, a remarkable type of
process whose current state encodes its whole history.
