  Compass models provide insights into the properties of Mott-insulating
materials that host bond-dependent anisotropic interactions between their
pseudospin degrees of freedom. In this article, we explore the classical and
quantum ground state properties of one such model relevant to certain layered
perovskite materials akin to Ba$_2$IrO$_4$ - namely, the Heisenberg-compass
model on the square lattice. We first investigate the ground state phase
diagram of this model using classical Monte Carlo simulations. These reveal
that the low temperature classical phase diagram is divided into six different
classes of long-range ordered phases, including four phases that exhibit an
order by disorder selection and two phases that are stabilized energetically.
This model admits a special duality transformation, known as the Klein duality,
conveniently allowing to map one region of coupling parameters onto another and
constraining the phase diagram, and which we exploit in our study. From the
analysis of the zero-point energy and the free energy of the spin waves, we
find that order by quantum disorder at zero temperature and order by thermal
disorder select the same orderings as those found from classical Monte Carlo
simulations. We further investigate the quantum ground states of this model
using numerical exact diagonalization on small clusters by exploiting the
translational symmetry of the square lattice. We obtain a ground state phase
diagram bearing close resemblance to that found from the classical analysis.
