When two hands require different information in bimanual asymmetric movements, interference can occur via callosal connections and ipsilateral corticospinal pathways. This interference could potentially work as a cost-effective measure in symmetric movements, allowing the same information to be commonly available to both hands at once. Using functional magnetic resonance imaging, we investigated supra-additive and sub-additive neural interactions in bimanual movements during the initiation and continuation phases of movement. We compared activity during bimanual asymmetric and symmetric movements with the sum of activity during unimanual right and left finger-tapping. Supra-additive continuation-related activation was found in the right dorsal premotor cortex and left cerebellum (lobule V) during asymmetric movements. In addition, for unimanual movements, the right dorsal premotor cortex and left cerebellum (lobule V) showed significant activation only for left-hand (non-dominant) movements, but not for right-hand movements. These results suggest that resource-demanding interactions in bimanual asymmetric movements are involved in a non-dominant hand motor network that functions to keep non-dominant hand movements stable. We found sub-additive continuation-related activation in the supplementary motor area (SMA), bilateral cerebellum (lobule VI) in symmetric movements, and the SMA in asymmetric movements. This suggests that no extra demands were placed on these areas in bimanual movements despite the conventional notion that they play crucial roles in bimanual coordination. Sub-additive initiation-related activation in the left anterior putamen suggests that symmetric movements place lower demands on motor programming. These findings indicate that, depending on coordination patterns, the neural substrates of bimanual movements either exhibit greater effort to keep non-dominant hand movements stable, or save neural cost by sharing information commonly to both hands.