Speech-associated gestures are hand and arm movements that not only convey semantic information to listeners but are themselves actions. Broca's area has been assumed to play an important role both in semantic retrieval or selection (as part of a language comprehension system) and in action recognition (as part of a "mirror" or "observation-execution matching" system). We asked whether the role that Broca's area plays in processing speech-associated gestures is consistent with the semantic retrieval/selection account (predicting relatively weak interactions between Broca's area and other cortical areas because the meaningful information that speech-associated gestures convey reduces semantic ambiguity and thus reduces the need for semantic retrieval/selection) or the action recognition account (predicting strong interactions between Broca's area and other cortical areas because speech-associated gestures are goal-direct actions that are "mirrored"). We compared the functional connectivity of Broca's area with other cortical areas when participants listened to stories while watching meaningful speech-associated gestures, speech-irrelevant self-grooming hand movements, or no hand movements. A network analysis of neuroimaging data showed that interactions involving Broca's area and other cortical areas were "weakest" when spoken language was accompanied by meaningful speech-associated gestures, and "strongest" when spoken language was accompanied by self-grooming hand movements or by no hand movements at all. Results are discussed with respect to the role that the human mirror system plays in processing speech-associated movements.