The canonical frameworks of viral evolution describe viruses as cellular predecessors, reduced forms of cells, or entities that escaped cellular control. The discovery of giant viruses has changed these standard paradigms. Their genetic, proteomic and structural complexities resemble those of cells, prompting a redefinition and reclassification of viruses. In a previous genome-wide analysis of the evolution of structural domains in proteomes, with domains defined at the fold superfamily level, we found the origins of viruses intertwined with those of ancient cells. Here, we extend these data-driven analyses to the study of fold families confirming the co-evolution of viruses and ancient cells and the genetic ability of viruses to foster molecular innovation. The results support our suggestion that viruses arose by genomic reduction from ancient cells and validate a co-evolutionary 'symbiogenic' model of viral origins.