The method of “antithetic variates” for Monte Carlo sampling was invented and named by Hammersley and Morton in 1956 and has been generalized by Halton and Handscomb, and Laurent. Given only that a Monte Carlo estimator possesses derivatives up to a certain order, in the sample space, transformations of the estimator are supplied (independent of the particular estimator used), which reduce the variance of the resulting estimates in a very marked degree. The explicit forms of these transformations are derived. It is demonstrated that, contrary to common belief, the transformations of Halton and Handscomb are more efficient than those proposed by Laurent.