The study of brain asymmetries, that were once believed to be a typically human trait subserving linguistic functions and handedness, has recently undergone a dramatic transformation. The knowledge that was gathered until a few decades ago by human neuropsychology alone became greatly complemented and extended by data from comparative psychology, neuroscience, and developmental biology. This contribution provided a more complete picture of the phylogenetic, ontogenetic, and ecological factors that confer advantages to organisms exploiting neural and cognitive lateralization as a general biological trait. Game theory and population genetics contribute to the understanding of these advantages in terms of alignment of interactions between individuals, at a social and communicative level. A comparative approach, combining ethological, developmental, and game-theoretical research will investigate the co-evolution of neural and morphological lateralization with social behaviours, and its impact on the evolution of higher cognitive abilities like language. The proposed research program will intensify the collaboration between insofar-isolated research domains, creating reciprocity between the biological and the behavioural sciences around a topic that is revealing a most promising tool for the understanding of the origins of the human mind.