We study the ground state ordering and interactions between two two-dimensional Wigner crystals on neutralizing charged plates by means of computer simulation. We consider crystals formed by i) point-like charges and ii) charged dimers, which mimic the screening of charged surfaces by elongated multivalent ions such as aspherical globular proteins, charged dendrimers or short stiff polyelectrolytes. Both systems, with point-like and dimeric ions, display five distinct crystalline phases on increasing the interlayer distance. In addition to alteration of translational ordering within the bilayer, the phase transitions in the dimeric system are characterized by alteration of orientational ordering of the ions.