  Pixelizations of Platonic solids such as the cube and icosahedron have been
widely used to represent spherical data, from climate records to Cosmic
Microwave Background maps. Platonic solids have well-known global symmetries.
Once we pixelize each face of the solid, each face also possesses its own local
symmetries in the form of Euclidean isometries. One way to combine these
symmetries is through a hierarchy. However, this approach does not adequately
model the interplay between the two levels of symmetry transformations. We show
how to model this interplay using ideas from group theory, identify the
equivariant linear maps, and introduce equivariant padding that respects these
symmetries. Deep networks that use these maps as their building blocks
generalize gauge equivariant CNNs on pixelized spheres. These deep networks
achieve state-of-the-art results on semantic segmentation for climate data and
omnidirectional image processing. Code is available at https://git.io/JGiZA.
