- Amorphous solids, as well as many other disordered materials, display remarkable universality in their low temperature acoustic properties. This universality is attributed to the attenuation of phonons by tunneling two-level systems (TLSs), facilitated by the interaction of the TLSs with the phonon field. Recently a Two-TLS model was introduced, where it was shown that inversion symmetric TLSs, i.e. TLSs having their two states related by local inversion symmetry, dictate phonon attenuation at low temperatures, and are responsible for the universal phenomena. The Two-TLS model suggests resolution to some long standing questions regarding the universal phenomena, such as the smallness and universality of the tunneling strength, and the energy scale of 3K below which universality is observed. A crucial part of the Two-TLS model lies in the magnitude and spatial dependence of effective TLS-TLS interactions involving inversion symmetric TLSs. Here we calculate numerically, using conjugate gradients method, the effective elastic TLS-TLS interactions for inversion symmetric and asymmetric TLSs, for pure and disordered solids, in two and three dimensions. We verify analytical estimates used in the Two-TLS model. We further show that the disorder induced power reduction of the spatial dependence of the interaction persists to short distances, not much larger than the interatomic distance. Our results provide strong support to the Two-TLS model, and contribute to the better understanding of the microscopic structure of amorphous solids and the low temperature properties of glasses.