  We develop a new method of extracting simultaneous measurements of weak
lensing shear and a local rotation of the plane of polarization using
observations of resolved radio sources. We show that the direction of
polarization is statistically linked with that of the gradient of the total
intensity field, and this provides the basis of our method. Using a number of
sources spread over the sky, this method allows constraints to be placed on
cosmic shear and birefringence, and it can be applied to any resolved radio
sources for which such a correlation exists. Assuming that the rotation and
shear are constant across the source, we use this relationship to construct a
quadratic estimator and investigate its properties using simulated
observations. We develop a calibration scheme using simulations based on the
observed images to mitigate a bias which occurs in the presence of measurement
errors and an astrophysical scatter on the polarization. The method is applied
directly to archival data of radio galaxies where we measure a mean rotation
signal of $\left<\omega\right>=-2.03^{\circ}\pm0.75^{\circ}$ and an average
shear compatible with zero using 30 reliable sources. This level of constraint
on an overall rotation is comparable with current leading constraints from CMB
experiments and is expected to increase by at least an order of magnitude with
future high precision radio surveys, such as those performed by the SKA. We
also measure the shear and rotation two-point correlation functions and
estimate the number of sources required to detect shear and rotation
correlations in future surveys.
