In the two-treatment, two-period crossover trial, patients are randomly allocated either to one group that receives treatment A followed by treatment B, or to another group that receives the treatments in the reverse order. Grizzle first proposed a two-stage procedure for analysing the data from such a trial. This paper examines the long-run sampling properties of this procedure, in terms of mean square error of point estimates, coverage probability of confidence intervals and actual significance level of hypothesis tests for the differences between the effects of the two treatments. The advantages of incorporating baseline observations into the analysis are also explored. Because the preliminary test for carryover is highly correlated with the analysis of data from the first period only, actual significance levels are higher than nominal levels even when there is no differential carryover. When carryover is present, the nominal level very seriously understates the actual level, and this becomes even worse when baseline observations are ignored. Increasing sample size only exacerbates the problem since this adverse behaviour then occurs at smaller values of the carryover effect. It is concluded that the two-stage analysis is too potentially misleading to be of practical use.