  We study effects of the spontaneous symmetry-breaking (SSB) in solitons built
of the dipolar Bose-Einstein condensate (BEC), trapped in a dual-core system
with the dipole-dipole interactions (DDIs) and hopping between the cores. Two
realizations of such a matter-wave coupler are introduced, weakly- and
strongly-coupled. The former one in based on two parallel pipe-shaped traps,
while the latter one is represented by a single pipe sliced by an external
field into parallel layers. The dipoles are oriented along axes of the pipes.
In these systems, the dual-core solitons feature the SSB of the supercritical
type and subcritical types, respectively. Stability regions are identified for
symmetric and asymmetric solitons, and, in addition, for non-bifurcating
antisymmetric ones, as well as for symmetric flat states, which may also be
stable in the strongly-coupled system, due to competition between the
attractive and repulsive intra- and inter-core DDIs. Effects of the contact
interactions are considered too. Collisions between moving asymmetric solitons
in the weakly-symmetric system feature elastic rebound, merger into a single
breather, and passage accompanied by excitation of intrinsic vibrations of the
solitons, for small, intermediate, and large collision velocities,
respectively. A PT-symmetric version of the weakly-coupled system is briefly
considered too, which may be relevant for matter-wave lasers. Stability
boundaries for PT-symmetric and antisymmetric solitons are identified.
