We describe a general framework for systematically treating halos that are due to refraction in preferentially oriented ice wedges, and we construct an atlas of such halos. The atlas is thus a very general collection of refraction halos that includes known halos as a small fraction. Each halo in the atlas is characterized by three parameters: the wedge angle, the zenith angle of the spin vector, and the spin vector expressed in the wedge frame. The theory reveals order in what seems initially to be a staggering variety of halo shapes, and in particular it explains why halos look the way they do.