Today there is a wealth of fascinating studies of connexive logical systems. But sometimes it looks as if connexive logic is still in search of a convincing interpretation that explains in intuitive terms why the connexive principles should be valid. In this paper I argue that difference-making conditionals as presented in Rott (Review of Symbolic Logic 15, 2022) offer one principled way of interpreting connexive principles. From a philosophical point of view, the idea of difference-making demands full, unrestricted connexivity, because neither logical truths nor contradictions or other absurdities can ever 'make a difference' (i.e., be relevantly connected) to anything. However, difference-making conditionals have so far been only partially connexive. I show how the existing analysis of difference-making conditionals can be reshaped to obtain full connexivity. The classical AGM belief revision model is replaced by a conceivability-limited revision model that serves as the semantic base for the analysis. The key point of the latter is that the agent should never accept any absurdities.