We consider a dipolar Bose-Einstein condensate trapped in an array of two-well systems with an arbitrary orientation 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 modeled by a pair of coupled discrete Gross-Pitaevskii equations, with dipole-dipole self-interactions 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, viz., on-site- and off-site-centered solitons, 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 PT-symmetric version of the system are constructed too, and stability areas are identified for them.