In social interactions, each individual's brain drives an action that, in turn, elicits systematic neural responses in their partner that drive a reaction. Consequently, the brain responses of both interactants become temporally contingent upon one another through the actions they generate, and different interaction dynamics will be underpinned by distinct forms of between-brain coupling. In this study, we investigated this by "performing functional magnetic resonance imaging on two individuals simultaneously (dual-fMRI) while they competed or cooperated with one another in a turn-based or concurrent fashion." To assess whether distinct patterns of neural coupling were associated with these different interactions, we combined two data-driven, model-free analytical techniques: group-independent component analysis and inter-subject correlation. This revealed four distinct patterns of brain responses that were temporally aligned between interactants: one emerged during co-operative exchanges and encompassed brain regions involved in social cognitive processing, such as the temporo-parietal cortex. The other three were associated with competitive exchanges and comprised brain systems implicated in visuo-motor processing and social decision-making, including the cerebellum and anterior cingulate cortex. Interestingly, neural coupling was significantly stronger in concurrent relative to turn-based exchanges. These results demonstrate the utility of data-driven approaches applied to "dual-fMRI" data in elucidating the interpersonal neural processes that give rise to the two-in-one dynamic characterizing social interaction.