arXiv:2507.16370v1 Announce Type: new Abstract: Counterfactual reasoning aims at answering contrary-to-fact questions like ''Would have Alice recovered had she taken aspirin?'' and corresponds to the most fine-grained layer of causation. Critically, while many counterfactual statements cannot be falsified -- even by randomized experiments -- they underpin fundamental concepts like individual-wise fairness. Therefore, providing models to formalize and implement counterfactual beliefs remains a fundamental scientific problem. In the Markovian setting of Pearl's causal framework, we propose an alternative approach to structural causal models to represent counterfactuals compatible with a given causal graphical model. More precisely, we introduce counterfactual models, also called canonical representations of structural causal models. They enable analysts to choose a counterfactual conception via random-process probability distributions with preassigned marginals and characterize the counterfactual equivalence class of structural causal models. Then, we present a normalization procedure to describe and implement various counterfactual conceptions. Compared to structural causal models, it allows to specify many counterfactual conceptions without altering the observational and interventional constraints. Moreover, the content of the model corresponding to the counterfactual layer does not need to be estimated; only to make a choice. Finally, we illustrate the specific role of counterfactuals in causality and the benefits of our approach on theoretical and numerical examples.