Many nonparametric competitors to Hotelling'sT2test lackT2's property of invariance under linear transformations. Such tests have power and efficiency that depend on the direction of shift and the covariance matrix of the alternative distribution. This article modifies a previously proposed bivariate sign test and signed rank test to overcome this difficulty. Asymptotic results and a Monte Carlo study indicate that the efficiency and small-sample power of the new procedures compare favorably to those of the original tests. An example illustrates the application of the new procedures.