The distortion field defined by the ellipticities of galaxy shapes as 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.