  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, e.g. for different spin values $S$ and number of
Kondo channels $N$. 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 weak-coupling RG treatment but can
be reproduced in brute-force perturbation theory. Employing numerical
renormalization group (NRG), we show that for an under-screened moment ($S=1$,
$N=1$) with easy-plane anisotropy, a crossing of magnetic susceptibilities can
also occur in the strong-coupling regime (below the Kondo temperature). This
suggests that collective magnetic ordering of such under-screened moments would
develop along the magnetic hard axis.
