  While gravitational lens inversion holds great promise to reveal the
structure of the light-deflecting mass distribution, both light and dark, the
existence of various kinds of degeneracies implies that care must be taken when
interpreting the resulting lens models. This article illustrates how thinking
in terms of the projected potential helps to gain insight into these matters.
Additionally it is shown explicitly how, when starting from a discretised
version of the projected potential of one particular lens model, the technique
of quadratic programming can be used to create a multitude of equivalent lens
models that preserve all or a subset of lens properties. This method is applied
to a number of scenarios, showing the lack of grasp on the mass outside the
strong lensing region, revisiting mass redistribution in between images and
applying this to a recent model of the SDSS J1004+4112 cluster, as well as
illustrating the generalised mass sheet degeneracy and source-position
transformation. In the case of J1004 we show that this mass redistribution did
not succeed at completely eliminating a dark mass clump recovered by GRALE near
one of the quasar images.
