Gauge invariance is invariance under local changes of gauge representative. Gauge-invariant observables depend only on the physical gauge orbit.
A gauge theory uses this redundancy to describe the same physical state by different field representatives.
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 transformation compensates. For a hedgehog ansatz for a monopole, a spatial rotation is compensated by the corresponding internal adjoint rotation. Gauge-invariant densities remain invariant without choosing a special gauge.
A local field combination is a gauge-invariant operator when its gauge representations and charges combine to a singlet. In an effective field theory, such operators may be multiplied by Wilson coefficients and inverse powers of a heavy scale. Multiplication by preserves the invariance of an existing Higgs Yukawa operator.
Gauge redundancy means that multiple field configurations or polarization representatives describe the same physical state. Physical amplitudes must be unchanged when an external polarization is shifted by a pure-gauge component.
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