This work poses the problem of estimating traffic signal phases from a sequence of maneuvers recorded from a turning movement counter. Inspired by the part-of-speech tagging problem in natural language processing, a hidden Markov model of the intersection is proposed. The model is calibrated from maneuver observations using the Baum-Welch algorithm, and the trained model is used to infer phases via the Viterbi algorithm. The approach is validated through numerical and experimental tests, which highlight that good performance can be achieved when sufficient training data is available, and when diverse maneuvers are observed during each phase. The supporting codes and data are available to download at https://github.com/reisiga2/Estimating-phases-from-turning-movement-counts.