Semantic categories, and the concepts belonging to them, have commonly been defined by their relative concreteness, that is, their reliance on perception. However, sensorimotor grounding must be regarded as going beyond the basic five senses and incorporate a multi-dimensional variety of perceptual and action experience. We present a series of exploratory analyses examining the sensorimotor grounding of participant-produced member concepts for 117 categories, spanning concrete (e.g. animal and furniture) and highly abstract (e.g. unit of time and science) categories. We found that both concrete and abstract categories are strongly grounded in multi-dimensional sensorimotor experience. Both domains were dominated by vision and, to a lesser extent, head movements, but concrete categories were more grounded in touch and hand–arm action, while abstract categories were more grounded in hearing and interoception. Importantly, this pattern of grounding was not uniform, and subdomains of concrete (e.g. ingestibles, animates, natural categories and artefacts) and abstract (e.g. internal, social and non-social) categories were grounded in different profiles of sensorimotor experience. Overall, these findings suggest that the distinction between abstract and concrete categories is not as clearcut as ontological assumptions might suggest, and that the strength and diversity of sensorimotor grounding in abstract categories must not be underestimated. This article is part of the theme issue ‘Concepts in interaction: social engagement and inner experiences’.