In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters.