  When a dyon encircles an infinitely-long solenoid enclosing uniform electric
and magnetic fields, its wave function accumulates a duality-invariant quantum
phase, which is topological because it depends on a winding number and is
nonlocal because the enclosed fields act on the dyon in regions where these
fields vanish. Here, we derive this dyon phase and show how its duality
symmetry unifies the Aharonov-Bohm phase with its dual phase. We obtain the
energy levels, the two-slit interference shift, and the scattering amplitude
associated with the duality-invariant quantum phase. Assuming that the dyon has
spin 1/2, we show that this spin does not affect the introduced phase. We argue
that a spin 1/2 dyon has electric and magnetic moments, the former being
greater than the latter because of the Schwinger-Zwanziger quantisation
condition.
