The comparison of primary interest in a 2 × 2 crossover trial typically concerns the effect of the treatments, sayAandB, on the mean response level. This article deals with another important aspect, namely the within-subject response variability underAandB. Differences in drug formulation and/or administration may lead to considerable differences in within-subject variability, whatever the difference in terms of mean level. Consideration of both these aspects may, therefore, be of considerable importance for the judgement of the treatments. It is shown that although there are no within-subject treatment replications, it is possible to make various exact inferences about theA/Bratio of within-subject variances and about the (A – B)-difference in mean level, simultaneously and marginally. These inferences are semiparametric in that no distributional assumption is made about the between-subject variability, whereas a normality assumption is used for the within-subject variability. The inferences include tests, confidence regions, and a multiple test procedure. A power approximation is also given. The results are illustrated numerically.