Although left-hemisphere (LH) specialization for language is often viewed as a key example of functional lateralization, there is increasing evidence that the right hemisphere (RH) can also extract meaning from words and sentences. However, the right hemisphere’s ability to appreciate syntactic aspects of language remains poorly understood. In the current study, we used separable, functionally well-characterized electrophysiological indices of lexico-semantic and syntactic processes to demonstrate RH sensitivity to syntactic violations among right-handers with a strong manual preference. Critically, however, the nature of this RH sensitivity to structural information was modulated by a genetically determined factor—familial sinistrality. The right hemisphere in right-handers without left-handed family members processed syntactic violations via the words’ accompanying lexico-semantic unexpectedness. In contrast, the right hemisphere in right-handers with left-handed family members could process syntactic information in a manner qualitatively similar to that of the left hemisphere