This paper introduces two new concepts of symmetry for multivariate copulas with a focus on tails regions. Properties of the symmetry concepts are investigated for bivariate copulas and a connection to radial symmetry is established. Two nonparametric testing procedures for the new concepts are developed using a vector of locally most powerful rank test statistics, applied to a new generalization of the FGM copula which parameterizes every vertex of the unit cube. This vector quantities deviations from independence in each vertex and the tests for the new symmetry concepts are based on comparisons of these deviations. It is shown that one of the new tests can also be used to test for radial symmetry, which results in a similar power of detecting bivariate radial symmetry compared to recently published nonparametric tests. Further, an application to insurance data is provided. Finally, an improvement of the selection process in the context of vine copula fitting is proposed that is based on the elimination of copula families with unsuitable symmetry properties.