Theories of grounded and embodied cognition offer a range of accounts of how reasoning and body-based processes are related to each other. To advance theories of grounded and embodied cognition, we explore the "cognitive relevance" of particular body states to associated math concepts. We test competing models of action-cognition transduction to investigate the cognitive relevance of directed actions to students' mathematical reasoning in the area of geometry. The hypotheses we test include (1) that cognitively relevant directed actions have a direct effect on performance (direct cognitive relevance hypothesis), (2) that cognitively relevant directed actions lead to more frequent production of gestures during explanations, which leads to improved performance (mediated cognitive relevance hypothesis), and (3) that performance effects of directed actions are influenced by the presence or absence of gesture production during mathematical explanations (moderated cognitive relevance hypothesis). We explore these hypotheses in an experiment where high school students (N = 85) evaluated the truth of geometry conjectures after performing cognitively relevant or cognitively irrelevant directed actions while playing a movement-based video game. Contrary to the direct and mediated cognitive relevance hypotheses, we found no overall differences in performance or gesture production between relevant and irrelevant conditions. Consistent with the moderated cognitive relevance hypothesis, cognitive relevance influenced mathematical performance, as measured by the accuracy of students' intuitions, insights, and the validity of their proofs, provided that students produced certain kinds of gestures during mathematical explanations (i.e., with explanatory gestures as the moderator). Implications for theories of grounded and embodied cognition and the design of embodied forms of educational interventions are discussed.