In a quadruply imaged lens system, the angular distribution of images around the lens centre is completely described by three relative angles. We show empirically that in the three-dimensional space of these angles, spanning 180°× 180°× 90°, quads from simple twofold symmetric lenses of arbitrary radial density profile and arbitrary radially dependent ellipticity or external shear define a nearly invariant two-dimensional surface. We give a fitting formula for the surface using the SIS+elliptical lensing potential. Various circularly symmetric mass distributions with shear up to γ∼ 0.4 deviate from it by typically an rmsof ∼01, while elliptical mass distributions with ellipticity of up to e∼ 0.4 deviate from it by rms ∼15. The existence of a near-invariant surface gives a new insight into the lensing theory and provides a framework for studying quads. It also allows one to gain information about the lens mass distribution from the image positions alone, without any recourse to mass modelling. As an illustration, we show that about 3/4 of observed galaxy-lens quads do not belong to this surface within observational error, and so require additional external shear or substructure to be modelled adequately.