  The distortion field defined by the ellipticities of galaxy shapes projected
on the sky can be uniquely decomposed into a gradient and a curl component. If
the observed ellipticities are induced by weak gravitational lensing, then the
distortion field is curl free. Here we show that, in contrast, the distortion
field resulting from intrinsic spin alignments is not curl free. This provides
a powerful discriminant between lensing and intrinsic contributions to observed
ellipticity correlations. We also show how these contributions can be
disentangled statistically from the ellipticity correlations or computed
locally from circular integrals of the ellipticity field. This allows for an
unambiguous detection of intrinsic galaxy alignments in the data. When the
distortions are dominated by lensing, as occurs at high redshifts, the
decomposition provides a valuable tool for understanding properties of the
noise and systematic errors. These techniques can be applied equally well to
the polarization of the microwave background, where it can be used to separate
curl-free scalar perturbations from those produced by gravity waves or defects.
