We argue that the grammatical diversity observed among the world's languages emerges from the struggle between individual cognitive systems trying to impose their preferred structure on human language. We investigate the cognitive bases of the two most common word orders in the world's languages: SOV (Subject-Object-Verb) and SVO. Evidence from language change, grammaticalization, stability of order, and theoretical arguments, indicates a syntactic preference for SVO. The reason for the prominence of SOV languages is not as clear. In two gesture-production experiments and one gesture comprehension experiment, we show that SOV emerges as the preferred constituent configuration in participants whose native languages (Italian and Turkish) have different word orders. We propose that improvised communication does not rely on the computational system of grammar. The results of a fourth experiment, where participants comprehended strings of prosodically flat words in their native language, shows that the computational system of grammar prefers the orthogonal Verb-Object orders. (Contains 5 figures.)