  Based on the standard gravitational lensing formalism with its effective,
projected lensing potential in a given background cosmology, we investigate
under which transformations of the source position and of the deflection angle
the observable properties of the multiple images, i.e. the time delay
differences, the relative image positions, relative shapes, and magnification
ratios, remain invariant. As these observables only constrain local lens
properties, we derive general, local invariance transformations in the areas
covered by the multiple images. We show that the known global invariance
transformations, e.g. the mass sheet transformation or the source position
transformation, are contained in our invariance transformations, when they are
restricted to the areas covered by the multiple images and when
lens-model-based degeneracies are ignored, like the freedom to add or subtract
masses in unconstrained regions without multiple images. Hence, we have
identified the general class of invariance transformations that can occur, in
particular in our model-independent local characterisation of strong
gravitational lenses.
