  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.
