In finance, random quantities such as returns frequently exhibit asymmetry in their tail behavior. The tail behavior is described using an elliptical distribution owing to its heaviness. Moreover, elliptical distributions constitute a parametric model class of multivariate regular variations, and the scale matrix is important in determining the tail dependence. For testing the asymmetry of the tail dependence, we propose a likelihood ratio test that checks the homogeneity of the scale matrix in separate regions. Moreover, the proposed test is applied to a real dataset to investigate the asymmetry of tail dependence.