This paper discusses the application of multivariate Mann-Whitney estimators to the comparison of two treatments for a strictly ordinal response variable in a crossover study with four sequence groups and three periods. Ways of managing randomly missing data and nonparametric covariance adjustment for no differences among groups for a baseline period have consideration as well. Estimators pertaining to treatment comparisons in linear logistic models for the Mann-Whitney estimators have determination through a Bradley-Terry model for dimension reduction and weighted least squares. These estimators can be the basis for both statistical tests and confidence intervals. The methods in this paper have their results presented for an example. Simulation studies for the methods show that they have reasonable control of type 1 error and power.