Crossover designs are well known to have major advantages when comparing the effect of two treatments which do not interact. With a right-censored survival endpoint, however, this design is quickly abandoned in favour of the more costly parallel design. Motivated by human immunodeficiency virus (HIV) prevention studies which lacked power, we evaluate what may be gained in this setting and compare parallel with crossover designs. In a heterogeneous population, we find and explain a substantial increase in power for the crossover study using a non-parametric logrank test. With frailties in a proportional hazards model, crossover designs equally lead to substantially smaller variance for the subject-specific hazard ratio (HR), while the population-averaged HR sees negligible gain. Its efficiency benefit is recovered when the population-averaged HR is reconstructed from estimated subject-specific hazard rates. We derive the time point for treatment crossover that optimizes efficiency and end with the analysis of two recent HIV prevention trials. We find that a Cellulose sulphate trial could have hardly gained efficiency from a crossover design, while a Nonoxynol-9 trial stood to gain substantial power. We conclude that there is a role for effective crossover designs in important classes of survival problems.