In this paper, we present an approach to treat and model the processing (i.e. recognition and production) of communicative gestures as grounded motor cognition. We first review cognitive theories and neuropsychological studies on human motor cognition. On this basis, we propose a computational framework that connects the sensorimotor processing of hand gestures in representational structures of meaning (visuospatial imagery), other modalities (language), and communicative intentions. We present an implementation that enables an embodied virtual agent to engage in gesture-based interaction with a human user.