Though the notion of exchangeability has been discussed in the causal inference literature under various guises, it has rarely taken its original meaning as a symmetry property of probability distributions. As this property is a standard component of Bayesian inference, we argue that in Bayesian causal inference it is natural to link the causal model, including the notion of confounding and definition of causal contrasts of interest, to the concept of exchangeability. Here we propose a probabilistic between-group exchangeability property as an identifying condition for causal effects, relate it to alternative conditions for unconfounded inferences (commonly stated using potential outcomes) and define causal contrasts in the presence of exchangeability in terms of posterior predictive expectations for further exchangeable units. While our main focus is on a point treatment setting, we also investigate how this reasoning carries over to longitudinal settings.