  We have recently (Heras et al. in Eur. Phys. J. Plus 136:847, 2021) argued
that classical electrodynamics can predict nonlocal effects by showing an
example of a topological and nonlocal electromagnetic angular momentum. In this
paper we discuss the dual of this angular momentum which is also topological
and nonlocal. We then unify both angular momenta by means of the
electromagnetic angular momentum arising in the configuration formed by a dyon
encircling an infinitely-long dual solenoid enclosing uniform electric and
magnetic fluxes and show that this electromagnetic angular momentum is
topological because it depends on a winding number, is nonlocal because the
electric and magnetic fields of this dual solenoid act on the dyon in regions
for which these fields are excluded and is invariant under electromagnetic
duality transformations. We explicitly verify that this duality-invariant
electromagnetic angular momentum is insensitive to the radiative effects of the
Li\'enard-Wiechert fields of the encircling dyon. We also show how duality
symmetry of this angular momentum suggests different physical interpretations
for the corresponding angular momenta that it unifies.
