Gravitational lensing deflects light. A single lens deflector can only shear images, but cannot induce rotations. Multiple lens planes can induce rotations. Such rotations can be observed in quadruply imaged sources, and can be used to distinguish between two proposed solutions of the flux anomaly problem: substructures in lensing galaxies vs large scale structure. We predict the expected amount of rotation due to large scale structure in strong lensing systems, and show how this effect can be measured using ~ mas VLBI astrometry of quadruple lenses with extended source structures. The magnitude of rotation is around one degree. The biggest theoretical uncertainty is the power spectrum of dark matter on very small scales. This procedure can potentially be turned around to measure the dark matter power spectrum on very small scales. We list the predicted RMS rotation angles for several quadruple lenses with known lens and source redshifts.

15 pages, 2 figures, corrected an error pointed out by Peter Schneider, now consistent with Schneider (1997), affects the results by 10-20%