  Understanding the interplay between topology and ordering in systems on
curved manifolds, governed by anisotropic interactions, takes a central role in
many fields of physics. In this paper, we investigate the effects of lattice
symmetry and local positional order on orientational ordering in systems of
long-range interacting point quadrupoles on a sphere in the zero temperature
limit. Locally triangular spherical lattices show long-range ordered
quadrupolar configurations only for specific symmetric lattices as strong
geometric frustration prevents general global ordering. Conversely, the ground
states on Caspar-Klug lattices are more diverse, with many different symmetries
depending on the position of quadrupoles within the fundamental domain. We also
show that by constraining the quadrupole tilts with respect to the surface
normal, which models interactions with the substrate, and by considering
general quadrupole tensors, we can manipulate the ground state configuration
symmetry.
