  We make some general remarks on long-ranged configurations in gauge or
diffeomorphism invariant theories where the fields are allowed to assume some
non vanishing values at spatial infinity. In this case the Gauss constraint
only eliminates those gauge degrees of freedom which lie in the connected
component of asymptotically trivial gauge transformations. This implies that
proper physical symmetries arise either from gauge transformations that reach
to infinity or those that are asymptotically trivial but do not lie in the
connected component of transformations within that class. The latter
transformations form a discrete subgroup of all symmetries whose position in
the ambient group has proven to have interesting implications. We explain this
for the dyon configuration in the $SO(3)$ Yang-Mills-Higgs theory, where we
prove that the asymptotic symmetry group is $Z_{|m|}\times \Re$ where $m$ is
the monopole number. We also discuss the application of the general setting to
general relativity and show that here the only implication of discrete
symmetries for the continuous part is a possible extension of the rotation
group $SO(3)$ to $SU(2)$.
