It is well known that closure is a necessary topological property for a reaction network to be dynamically stable. In this work we combine notions of chemical organization theory with structural properties of reaction networks to distill a minimal set of closed reaction networks that encodes the non-trivial stable dynamical regimes of the network. In particular, these non-trivial closed sets are synergetic, in the sense that their dynamics cannot always be computed from the dynamics of its closed constituents. We introduce a notion of separability for reaction networks and prove that it is strictly related to the notion of synergy. In particular, we provide a characterization of the non-trivial closed reaction networks by means of their degree of internal synergy. The less trivial the dynamics of the reaction network, the less can be separated into constituents, and equivalently the more synergies the reaction network has. We also discuss the computational and analytical benefits of this new representation of the dynamical structure of a reaction network.