 Symmetry up to gauge transformation

A spatial transformation is a symmetry of a gauge-field configuration when its pullback lies in the gauge orbit of the original configuration. It need not fix a particular representative. For a planar vortex, rotation changes its winding phase and a constant $U(1)$ transformation compensates. For a hedgehog ansatz for a monopole, a spatial rotation is compensated by the corresponding internal adjoint $SU(2)$ rotation. Gauge-invariant densities remain invariant without choosing a special gauge.