The circular polarization covariance matrix is a convenient method for expressing partially polarized response data with respect to a circularly polarized basis. However, little concerning either the properties of the circular polarization covariance matrix or methods for transforming data expressed in this format has been previously reported in the literature. Here we show (1) how to recover both the diagonal and off-diagonal elements of the circular polarization covariance matrix from response data stored in either Stokes matrix or linear polarization covariance matrix format, (2) how the contribution of physical scattering mechanisms such as odd-bounce, even-bounce, and diffuse or volume scattering are expressed in circular polarization covariance matrix format, and (3) the form of the response after rotation of the target about the radar line-of-sight. Next, we derive the constraints on the matrix elements (and thereby determine the dimensionality of the response) when a target exhibits reflection, rotation, azimuthal, or centrical symmetry. Because the circular polarimetric rotation operator has a particularly simple form, referring the polarization covariance matrix to a circularly polarized basis rather than a linearly polarized basis simplifies the formulation considerably. In many applications, circular polarimetric features are synthesized from data collected using a linear polarization diversity radar. We show that residual amplitude and phase imbalance between channels under a linear polarized basis transforms to cross-talk under the circularly polarized basis.