Estimating joint distributions (a.k.a. couplings) over counterfactual outcomes is central to personalized decision-making and treatment risk assessment. Two emergent frameworks with identifiability guarantees are: (i) bijective structural causal models (SCMs), which are flexible but brittle to mis-specified latent noise; and (ii) optimal-transport (OT) methods, which avoid latent noise assumptions but can produce incoherent counterfactual transports which fail to identify higher-order couplings. In this work, we bridge the gap with \emph{counterfactual cocycles}: a framework for counterfactual transports that use algebraic structure to provide coherence and identifiability guarantees. Every counterfactual cocycle corresponds to an equivalence class of SCMs, however the cocycle is invariant to the latent noise distribution, enabling us to sidestep various mis-specification problems. We characterize the structure of all identifiable counterfactual cocycles; propose flexible model parameterizations; introduce a novel cocycle estimator that avoids any distributional assumptions; and derive mis-specification robustness properties of the resulting counterfactual inference method. We demonstrate state-of-the-art performance and noise-robustness of counterfactual cocycles across synthetic benchmarks and a 401(k) eligibility study.