Rayner, John and Thas, Oliver, 2005, More Informative Testing For Bivariate Symmetry, Australian and New Zealand Journal of Statistics, 47(2), 211-217.
In testing for bivariate symmetry against arbitrary alternatives the well-known test developed by Bowker in 1948 is shown to be a score test, and to have useful components. These components are asymptotically independent and asymptotically have the standard normal distribution. Moreover they assess particular pairs of cells for symmetry. These components can also be used in a data analytic manner to complement a test for bivariate symmetry against ordered alternatives.