Summary: An antithetic variates method for the bootstrap is proposed and discussed. It is applicable quite generally to bias estimation, distribution function estimation and quantile estimation, for example. It is based on an `antithetic permutation' of the sample, which amounts to ranking values of a certain function of the data. Once this has been done, B uniform resampling operations may be immediately converted into 2B `effective' resampling operations, yielding greater statistical efficiency than 2B totally independent resampling operations. We show that antithetic resampling leads to positive nonnegligible gains in performance, for the same level of labour, when compared with ordinary uniform resampling.