We investigate the role that the atomic structure of grain boundaries in Cu has in the interaction with point defects produced during irradiation. We focus on three aspects of boundary-defect interaction: how defects interact with pristine boundaries, how boundaries modify defect production during collision cascades, and how defects interact with damaged boundaries. We find that there are generic features common to most boundaries, including biased absorption of interstitials over vacancies during collision cascades and strong interactions with vacancies for interstitial-loaded boundaries. However, we find that the magnitude of these behaviors depends strongly on the atomic structure of the boundary. In particular, the biased absorption is much stronger for a high-angle twist boundary and smallest for a more general twist-asymmetric tilt boundary. Further, the strength of boundary-defect interactions is also sensitive to the boundary structure. We conclude that the sink strength of grain boundaries for interacting with point defects is not an intrinsic property of the boundary but rather depends on the irradiation conditions through the absorbed defect content at the boundary.