  We identify common features, seen across a wide range of actinide and
lanthanide based heavy-fermion metals, that we associate with the universal
properties of the residual moments generated by Kondo hybridisation. The
features include the opposing anisotropies of the effective moments and
Curie-Weiss constants, the reorientation of the magnetic easy axis, the
hard-axis susceptibility maximum and the hard-axis metamagnetic transition. The
extension of large-$N$ Read-Newns theory to the underscreened Kondo lattice
allows us to study large magnetic fluctuations of residual local moments in the
presence of strong $f$-$d$ hybridisation. We derive an effective Hamiltonian
for the residual moments and show that competing single-ion and exchange
anisotropies are responsible for the easy-axis reorientation. We identify a
strong-coupling regime, where the temperature of the easy-axis reorientation is
{\it directly proportional} to the coherence temperature and find good
agreement across several Eu and U based heavy-fermion families, as well as a
weak-coupling regime, where the easy axis flips just before the onset of
magnetic order, as is seen in many Ce and Yb heavy-fermion compounds. We show
that the exchange anisotropy of the residual moments can also generate a
maximum in the hard-axis susceptibility, and find good agreement with UTe$_2$
at low pressures, suggesting that this material is particularly well described
by the single-channel $S=1$ model. Finally, we speculate on the possible
pairing interactions generated by the fluctuations of the residual moments.
