We present a generic mechanism that explains why many Kondo materials show magnetic ordering along directions that are not favoured by the crystal-field anisotropy. Using a renormalization-group (RG) analysis of single impurity Kondo models with single-ion anisotropy, we demonstrate that strong fluctuations above the Kondo temperature drive a moment re-orientation over a wide range of parameters. In tetragonal systems this can happen for both easy-plane or easy axis anisotropy. The characteristic crossing of magnetic susceptibilities is not an artefact of the RG treatment but can be reproduced in brute-force perturbation theory. Finally, we demonstrate that anisotropic fluctuations on the single-ion level and the fluctuation-induced anisotropy of RKKY interactions cooperate to drive magnetic hard-direction ordering.