Anisotropy effects can significantly control or modify the ground-state properties of magnetic systems. Yet the origin and the relative importance of the possible anisotropy terms are difficult to assess experimentally and often ambiguous. Here we propose a technique that allows a very direct distinction between single-ion and two-ion anisotropy effects. The method is based on high-resolution neutron spectroscopic investigations of magnetic cluster excitations. This is exemplified for manganese dimers and tetramers in the mixed compounds CsMn_xMg_1–xBr₃ (0.05 ≤ x ≤ 0.40). Our experiments provide evidence for a pronounced anisotropy of the order of 3% of the dominant bilinear exchange interaction, and the anisotropy is dominated by the single-ion term. The detailed characterization of magnetic cluster excitations offers a convenient way to unravel anisotropy effects in any magnetic material.