In the vast clade of animals known as the bilateria, cerebral and behavioral asymmetries emerge against the backdrop of bilateral symmetry, with a functional trade-off between the two. Asymmetries can lead to more efficient processing and packaging of internal structures, but at the expense of efficient adaptation to a natural world without systematic left-right bias. Asymmetries may arise through the fissioning of ancestral structures that are largely symmetrical, creating new circuits. In humans these may include asymmetrical adaptations to language and manufacture, and as one or other hemisphere gains dominance for functions that were previously represented bilaterally. This is best illustrated in the evolution of such functions as language and tool manufacture in humans, which may derive from the mirror-neuron system in primates, but similar principles probably apply to the many other asymmetries now evident in a wide range of animals. Asymmetries arise in largely independent manner with multi-genetic sources, rather than as a single over-riding principle.