In testing whether a treatment has an effect or not, the experimenter is often obliged to use the same subjects for control and treated groups. In such a case it is generally unrealistic to assume independence and one is led to tests of bivariate symmetry. The object of this paper is to show that bivariate symmetry is not equivalent to univariate symmetry; that there exists a feasible procedure different from the likelihood ratio test in the normal case; that there is no unique "natural" concept of rank; that all distribution-free (DF) procedures are based on permutations; and that optimal DF procedures for a simple alternative are based on permutations of the likelihood function.
"Bivariate Symmetry Tests: Parametric and Nonparametric." Ann. Math. Statist. 40 (1) 259 - 269, February, 1969. https://doi.org/10.1214/aoms/1177697821