  Scattering measurements with incident linearly polarized $\gamma$ rays
provide information on spins, parities, and $\gamma$-ray multipolarity mixing
coefficients, and, therefore, on the nuclear matrix elements involved in the
transitions. We present the general formalism for analyzing the observed
angular correlations. The expressions are used to compute three-dimensional
radiation patterns, which are important tools for optimizing experimental
setups. Frequently, $\gamma$-ray transitions can proceed via two
multipolarities that mix coherently. In such cases, the relative phases of the
nuclear matrix elements are important when comparing results from different
measurements. We discuss different phase conventions that have been used in the
literature and present their relationships. Finally, we propose a basic
experimental geometry consisting of detectors located at four different spatial
locations. For this geometry, we present the measured anisotropies of the
emitted $\gamma$ rays in graphical format as an aid in the data analysis.
