We develop a mathematical and interpretative foundation for the enterprise of decision-theoretic statistical causality (DT), which is a straightforward way of representing and addressing causal questions. DT reframes causal inference as "assisted decision-making", and aims to understand when, and how, I can make use of external data, typically observational, to help me solve a decision problem by taking advantage of assumed relationships between the data and my problem. The relationships embodied in any representation of a causal problem require deeper justification, which is necessarily context-dependent. Here we clarify the considerations needed to support applications of the DT methodology. Exchangeability considerations are used to structure the required relationships, and a distinction drawn between intention to treat and intervention to treat forms the basis for the enabling condition of "ignorability". We also show how the DT perspective unifies and sheds light on other popular formalisations of statistical causality, including potential responses and directed acyclic graphs.