  Nanopatterned magnetic thin films offer a platform for exploration of
tailored magnetic properties such as emergent long-range order. A prominent
example is artificial spin ice (ASI), where an arrangement of nanoscale
magnetic elements, acting as macrospins, interact via their dipolar fields. In
this study, we discuss the transition from antiferromagnetic (AF) to
ferromagnetic (FM) long-range order in a square lattice ASI, as the magnetic
elements are gradually rotated through 45{\deg} to a "pinwheel" configuration.
The AF$\unicode{x2013}$FM transition is observed experimentally using
synchrotron radiation x-ray spectromicroscopy and occurs for a critical
rotation angle of the nanomagnets, contingent upon the dipolar coupling
determined by their separation in the lattice. Large-scale magnetic dipole
simulations show that the point-dipole approximation fails to capture the
correct AF$\unicode{x2013}$FM transition angle. However, excellent agreement
with experimental data is obtained using a dumbbell-dipole model which better
reflects the actual dipolar fields of the magnets. This model explains the
coupling-dependence of the transition angle, another feature not captured by
the pointdipole model. Our findings resolve a discrepancy between measurement
and theory in previous work on "pinwheel" ASIs. Control of the
AF$\unicode{x2013}$FM transition and this revised model open for improved
design of magnetic order in nanostructured systems.
