We study a strategic game model on hypergraphs where players, modelled by nodes, try to coordinate or anti-coordinate their choices within certain groups of players, modelled by hyperedges. We show this model to be a strict generalisation of symmetric additively separable hedonic games to the hypergraph setting and that such games always have a pure Nash equilibrium, which can be computed in pseudo-polynomial time. Moreover, in the pure coordination setting, we show that a strong equilibrium exists and can be computed in polynomial time when the game possesses a certain acyclic structure.