  We consider a dipolar Bose-Einstein condensate trapped in an array of
two-well systems with an arbitrary orientations of the dipoles relative to the
system's axis. The system can be built as a chain of local traps sliced into
two parallel lattices by a repelling laser sheet. It is modelled by a pair of
coupled discrete Gross-Pitaevskii equations, with dipole-dipole self- and
cross-interactions. When the dipoles are not polarized perpendicular or
parallel to the lattice, the cross-interaction is asymmetric, replacing the
familiar symmetric two-component discrete solitons by two new species of
cross-symmetric ones, on-site- and off-site-centered, which are strongly
affected by the orientation of the dipoles and separation between the parallel
lattices. A very narrow region of intermediate asymmetric discrete solitons is
found at the boundary between the on- and off-site families. Two different
types of solitons in the $\mathcal{PT}$%-symmetric version of the system are
constructed too, and stability areas are identified for them.
