Summary: This paper closely examines HMMs in which all the hidden random variables are binary. Its main contributions are (1) a birational parametrization for every such HMM, with an explicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraic model membership test for every such HMM, and (3) minimal defining equations for the 4-node fully binary model, comprising 21 quadrics and 29 cubics, which were computed using Gröbner bases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) are rationally identifiable in the sense of L. García-Puente, S. Spielvogel and S. Sullivant[“Identifying causal effects with computer algebra”, in: Proceedings of the 26th Conference of Uncertainty in Artificial Intelligence. Corvallis, Oregon: AUAI Press. 193–200 (2010)], and each model’s Zariski closure is therefore a rational projective variety of dimension 5. Gröbner basis computations for the model and its graph are found to be considerably faster using these parameters. In the case of two hidden states, item (2) supersedes a previous algorithm of Schönhuth which is only generically defined, and the defining equations (3) yield new invariants for HMMs of all lengths \(\geq 4\). Such invariants have been used successfully in model selection problems in phylogenetics, and one can hope for similar applications in the case of HMMs.