Shells of various viruses and other closed packed structures with spherical topology exhibit icosahedral symmetry because the surface of a sphere cannot be tiled without defects, and icosahedral symmetry yields the most symmetric configuration with the minimum number of defects. Icosahedral symmetry is different from icosahedral-shaped structures, which include some large viruses, cationic-anionic vesicles, and fullerenes. We present a faceting mechanism of ionic shells into icosahedral shapes that breaks icosahedral symmetry resulting from different arrangements of the charged components among the facets. These self-organized ionic structures may favor the formation of flat domains on curved surfaces. We show that icosahedral shapes without rotational symmetry can have lower energy than spheres with icosahedral symmetry caused by preferred bending directions in the planar ionic lattice. The ability to create icosahedral shapes without icosahedral symmetry may lead to the design of new functional materials. The electrostatically driven faceting mechanism we present here suggests that we can design faceted polyhedra with diverse symmetries by coassembling oppositely charged molecules of different stoichiometric ratios.