We consider the symmetry properties of a general class of nonlocal population models
describing the aggregation and alignment of oriented objects in two dimensions. Such objects could
be at the level of molecules, cells, or whole organisms. We show that the underlying interaction
kernel is invariant under the so-called shift-twist action of the Euclidean group acting on the space
R2 × S1. This group action was previously studied within the context of a continuum model of
primary visual cortex. We use perturbation methods to solve the eigenvalue problem arising from
linearization about a homogeneous state, and then use equivariant bifurcation theory to identify the
various types of doubly periodic patterns that are expected to arise when the homogeneous state
becomes unstable. We thus establish that two distinct forms of spatio-angular o