This article proposes a Mantel-Haenszel-type estimator of an assumed common cumulative odds ratio in a proportional odds model for an ordinal response with several 2 x c contingency tables. It is useful, for instance, for comparing two treatments on an ordinal response for data from several centers when the data are highly sparse. The estimator has behavior similar to the Mantel-Haenszel estimator of a common odds ratio for several 2 x 2 tables. It is consistent under the ordinary asymptotic framework in which the number of tables is fixed and, unlike the maximum likelihood (ML) estimator, also under sparse asymptotics in which the number of tables grows with the sample size. Simulations reveal a considerable difference between it and the ML estimator when each table has few observations. Efficiency comparisons suggest that little efficiency loss occurs compared to the ML estimator when the data are not sparse. Tests and estimators are presented for detecting and handling heterogeneity in the odds ratios, and generalizations are available for stratified r x c contingency tables.