Animal evolution is often presented as a march toward complexity, with different living animal groups each representing grades of organization that arose through the progressive acquisition of complex traits. There are now many reasons to reject this classical hypothesis. Not only is it incompatible with recent phylogenetic analyses, but it is also an artifact of 'hidden biology', that is, blind spots to complex traits in non-model species. A new hypothesis of animal evolution, where many complex traits have been repeatedly gained and lost, is emerging. As we discuss here, key details of this new model hinge on a better understanding of the Porifera and Ctenophora, which have each been hypothesized to be sister to all other animals, but are poorly studied and often misrepresented.