  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 is difficult to assess experimentally and often
ambiguous. Here we propose a technique which 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 CsMnxMg1-xBr3 (0.05\leqx\leq0.40). 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.
