To understand how neural networks in the left (LH) and right (RH) cerebral hemispheres contribute to different aspects of language comprehension, in two experiments we measured event-related potentials (ERPs) as right-handed participants read sentences, some of which contained morphosyntactic and thematic role violations. Replicating prior work (Kuperberg et al., 2006), in Experiment 1 thematic role violations elicited both an N400 and a (semantic) P600 effect. Morphosyntactic violations elicited effects that differed as a function of participants’ familial sinistrality (the presence FS+ or absence FS- of a left-handed biological relative): FS+ participants showed a (syntactic) P600 effect whereas FS- participants showed a biphasic N400 and P600 response. To assess whether this difference reflects different underlying patterns of lateralization, in Experiment 2 target words were presented using visual half-field (VF) presentation. Indeed, for morphosyntactic violations, the FS- group elicited an asymmetric pattern, showing a P600 effect only with LH-biased presentation and an N400 effect in both VFs (cf. Lee and Federmeier, 2015). In contrast, FS+ participants showed a bilateral (N400-only) response pattern. This provides further evidence of FS-based differences in hemispheric contributions to syntactic processing. Strikingly, we found that, when lateralized, thematic role violations did not elicit a P600 effect, suggesting that this effect requires contributions from both hemispheres. The different response patterns for morphosyntactic and thematic role animacy violations across FS and VF also point to a processing difference in the comprehension mechanisms underlying the semantic and syntactic P600, which had heretofore been assumed to be variants of the same component. •Familial sinistrality (FS) influenced the brain's response to syntactic violations.•For FS- group, hemifield of presentation influenced processing of syntactic violations.•FS+ group showed symmetrical brain responses for syntactic/thematic violations.•Normal processing of thematic role violations seems to require both hemispheres.•Syntactic and semantic P600s have different eliciting conditions.