Interareal connectomes are whole-brain wiring diagrams of white-matter pathways. Recent studies have identified modules, hubs, module hierarchies and rich clubs as structural hallmarks of these wiring diagrams. An influential current theory postulates that connectome modules are adequately explained by evolutionary pressures for wiring economy, but that the other hallmarks are not explained by such pressures and are therefore less trivial. Here, we use constraint network models to test these postulates in current gold-standard vertebrate and invertebrate interareal-connectome reconstructions. We show that empirical wiring-cost constraints inadequately explain connectome module organization, and that simultaneous module and hub constraints induce the structural byproducts of hierarchies and rich clubs. These byproducts, known as spandrels in evolutionary biology, include the structural substrate of the default-mode network. Our results imply that currently standard connectome characterizations are based on circular analyses or double dipping, and we emphasize an integrative approach to future connectome analyses for avoiding such pitfalls.