Concurrency and causality can be expressed within a labelled transition system by exploiting reversibility of transitions. It is natural to ask what behavioural equivalences can be captured by bisimulations in the reversible setting. In this paper we work with keyed configuration structures and \(\textsf{CCS}{\textsf{K}}\) , establish an operational correspondence between the two models, and give definitions of hereditary history-preserving bisimulation and history-preserving bisimulation in both models. We then present several characterisation results for the two bisimulations in terms of previously proposed, as well as new, “reverse” bisimulations.