  Inhomogeneous charge distributions have important repercussions on
electrostatic interactions in systems of charged particles but are often
difficult to examine theoretically. We investigate how electrostatic
interactions are influenced by patchy charge distributions exhibiting certain
point group symmetries. We derive a general form of the electrostatic
interaction energy of two permeable, arbitrarily charged spherical shells in
the Debye-H\"uckel approximation and apply it to the case of particles with
icosahedral, octahedral, and tetrahedral inhomogeneous charge distributions. We
analyze in detail how charge distribution symmetry modifies the interaction
energy and find that local charge inhomogeneities reduce the repulsion of two
overall equally charged particles, while sufficient orientational variation in
the charge distribution can turn the minimum interaction energy into an
attraction. Additionally we show that larger patches and thus lower symmetries
and wave numbers result in bigger attraction given the same variation.
