Measurements of the longitudinal and transverse magnetization components (M(L),M(T)) were performed on a field-cooled thin-disk-like sample of disordered Ni76Mn24 as it was rotated in various fixed fields. Our experimental data were presented at low temperatures which are well below the spin-glass freezing temperature (almost-equal-to 116 K). The observed rotational behavior of the anisotropy is consistent with neither a purely directional character nor a unidirectional and uniaxial character in the assumption of a rigid rotation. We also take into account the possible incoherent domain rotations. Such a possibility also does not seem solely to explain this rotational behavior. However, the data are well described by the fact that the unidirectional anisotropy is not rigidly linked to the lattice along its initial direction, but it can be elastically rotated by the applied field. These experimental results are also in good agreement with a ''domain anisotropy model'' suggesting a multidomain configuration even in a field-cooled state, which affects the rotation of the anisotropy.