Theoretical accounts of syntax are broadly divided into lexicalist or construction-based viewpoints, where lexicalist traditions argue that a great deal of syntactic information is stored in lexical representations, while construction-based views argue for separate representations of multiword syntactic structures. Moreover, a strict autonomy between syntactic and semantic processing has been posited based on the grammatical well-formedness of non-sense sentences such as This round table is square. In this paper, we provide an overview of these competing conceptions of syntactic structure and the role of syntax in grammar. We review converging neuroimaging, electrophysiological, behavioral, electrocorticographic, and computational modeling evidence that challenge these views. In particular, we show that a temporal lobe ventral stream is crucial in processing phrases involving nouns and attributive adjectives, while a dorsal stream involving left parietal regions, including the angular gyrus, is crucial in processing constructions involving verbs and relational adjectives. We additionally support this interpretation by examining divergent pathways in the visual system for processing object information and event/spatial information, on the basis of integration across visual and auditory modalities. Our interpretation suggests that combinatorial operations which combine words into phrases cannot be isolated to a single anatomical location, as has been previously proposed—instead, it is an instantiation of a more general neural computation, one that is implemented across various brain regions and can be utilized in service of constructing linguistic phrases. Based on this orientation, we explore how abstract syntactic constructions, such as the transitive construction, both mirror and could emerge from semantics. These abstract construction representations are argued to be distinct from, and stored in regions functionally downstream from, lexical representations of verbs. Comprehension therefore involves the integration of both representations via feedforward and feedback connections. We implicate the IFG in communicating across the language network, including correctly integrating nominal phrases with the overall event representation and serving as one interface between processing streams. Overall, this approach accords more generally with conceptions of the development of cognitive systematicity, and further draws attention to a potential role for the medial temporal lobe in syntactic behaviors, often overlooked in current neurofunctional accounts of syntactic processing.