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The Cloude-Pottier radar polarimetry paradigm, which is understood as the spectral decomposition theorem of the target coherency matrix plus the classification technique based on the triad of parameters given by entropy, alpha angle, and anisotropy, has become a very well-established methodology for treating high-resolution polarimetric radar images, particularly those obtained with synthetic aperture radar (SAR) sensors. This methodology is revisited here from the standpoint of the coherence and polarization theory and their mutual relationship, in the light of the interest aroused once again after Wolf's article on this subject in 2003. Despite its success in terms of acceptance by the SAR community, the Cloude-Pottier paradigm relies on the arguable assumption that different scattering mechanisms can be separated by diagonalizing the aforementioned coherency matrix and then assigning each corresponding eigenvector to one of the independent scattering mechanisms. Our main statement in this paper is that the coherency matrix illustrates the behavior of the target from the point of view of polarization and not of full coherence, which would justify this assumption, even if only partially. Therefore, it is not rigorous to identify each eigenvector of this decomposition with a distinct scattering mechanism. Cloude and Pottier argue that the eigendecomposition of the coherency matrix outperforms other target decomposition theorems due to its uniqueness. It is also suggested that this very uniqueness, together with the orthogonality of the eigenvectors, supports the injective mapping between scattering mechanisms and eigenvectors that was first assumed based on statistical independence. With the aim of providing a comprehensive overview of the problem, we discuss some important concepts with regard to both field and target coherency matrices. In addition, we revise the concepts of entropy, alpha angle, and anisotropy as defined by the aforementioned authors, which are - - also a central part of this paradigm and which have played an important role in SAR image classification since its introduction. Again, some disagreement is found with the meaning of these parameters as they have been discussed so far.