Different forms of antagonism are classified in terms of response types and are related to the sufficient-cause framework. These forms of antagonism include "synergy under recoding of an exposure," "synergism under recoding of the outcome," and so-called "competing response types," with synergism itself conceived of as causal co-action within the sufficient-cause framework. In this paper, the authors show that subadditivity necessarily implies at least one of these 3 forms of antagonism. Empirical conditions for specific forms of antagonism are given for settings in which monotonicity assumptions are and are not considered plausible. The implications of subadditivity and superadditivity for causal co-action for either an outcome or its absence are characterized under various assumptions about monotonicity. A simple computational procedure is described for assessing whether any specific form of causal co-action can be detected for either an outcome or its absence for both cohort and case-control data. The results in this paper are illustrated by application to examples drawn from the existing literature on gene-gene and gene-environment interactions.