The structure and symmetry of viruses has been the subject of study since Crick and Watson in 1956, and there have been several complementary theories describing different aspects of the geometry of these complicated entities. Included here is a unified theory that relates the structure and sizes of the different viral components, from the capsomeres to the packaging of the genomic material, providing, through a set of structural constraints on viral structures, a new classification scheme for viral structures. Moreover, aspects of this theory also apply to fullerene structures in chemistry, showing that this symmetry principle is deeper than just biological in nature.