  Multiple image gravitational lens systems, and especially quads are
invaluable in determining the amount and distribution of mass in galaxies. This
is usually done by mass modeling using parametric or free-form methods. An
alternative way of extracting information about lens mass distribution is to
use lensing degeneracies and invariants. Where applicable, they allow one to
make conclusions about whole classes of lenses without model fitting. Here, we
use approximate, but observationally useful invariants formed by the three
relative polar angles of quad images around the lens center to show that many
smooth elliptical+shear lenses can reproduce the same set of quad image angles
within observational error. This result allows us to show in a model-free way
what the general class of smooth elliptical+shear lenses looks like in the
three dimensional (3D) space of image relative angles, and that this
distribution does not match that of the observed quads. We conclude that, even
though smooth elliptical+shear lenses can reproduce individual quads, they
cannot reproduce the quad population. What is likely needed is substructure,
with clump masses larger than those responsible for flux ratio anomalies in
quads, or luminous or dark nearby perturber galaxies.
