  We study the coupling between backward- and forward-propagating wave modes,
with the same group velocity, in a composite right/left-handed nonlinear
transmission line. Using an asymptotic multiscale expansion technique, we
derive a system of two coupled nonlinear Schr{\"o}dinger equations governing
the evolution of the envelopes of these modes. We show that this system
supports a variety of backward- and forward propagating vector solitons, of the
bright-bright, bright-dark and dark-bright type. Performing systematic
numerical simulations in the framework of the original lattice that models the
transmission line, we study the propagation properties of the derived vector
soliton solutions. We show that all types of the predicted solitons exist, but
differ on their robustness: only bright-bright solitons propagate undistorted
for long times, while the other types are less robust, featuring shorter
lifetimes. In all cases, our analytical predictions are in a very good
agreement with the results of the simulations, at least up to times of the
order of the solitons' lifetimes.
