A Cramer von-Mises type statistic is proposed for testing the symmetry of a continuous distribution function. Its asymptotic null distribution is found explicitly, and its asymptotic distribution under a sequence of local alternatives is described. A Monte Carlo study indicates that the asymptotic formulae are accurate for sample sizes as small as twenty.
"A Cramer Von-Mises Type Statistic for Testing Symmetry." Ann. Math. Statist. 43 (6) 2035 - 2038, December, 1972. https://doi.org/10.1214/aoms/1177690879