  We analyze the classical version of a plaquette orbital model that was
recently introduced and studied numerically by S. Wenzel and W. Janke. In this
model, edges of the square lattice are partitioned into $x$ and $z$-types that
alternate along both coordinate directions and thus arrange into a checkerboard
pattern of $x$ and $z$-plaquettes; classical O(2)-spins are then coupled
ferromagnetically via their first components over the $x$-edges and via their
second components over the $z$-edges. We prove from first principles that, at
sufficiently low temperatures, the model exhibits orientational long-range
order (OLRO) in one of the two principal lattice directions. Magnetic order is
precluded by the underlying symmetries. A similar set of results is inferred
also for quantum systems with large spin although the $\frac12$ instance
currently seems beyond the reach of rigorous methods. We point out that the
Ne\'el order in the plaquette energy distribution observed in numerical
simulations is an artefact of the OLRO and a judicious choice of the plaquette
energies. In particular, this order seems to disappear when the plaquette
energies are adjusted to vanish at the ground-state level. We also discuss the
specific role of the underlying symmetries in Wenzel and Janke's simulations
and propose an enhanced method of numerical sampling that could in principle
significantly increase the speed of convergence.
