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Interferometric synthetic aperture radar (InSAR) is a phase-based radar signal processing technique that has been addressed from a polarimetric point of view since the late 1990s, starting with Cloude and Papathanassiou's foundational work. Polarimeric InSAR (PolInSAR) has consolidated as an active field of research in parallel to non-PolInSAR. Regarding the latter, there have been a number of issues that were discussed in an earlier paper from which some other questions related to Cloude's PolInSAR come out naturally. In particular, they affect the usual understanding of coherence and statistical independence. Coherence involves the behavior of electromagnetic waves in at least a pair of points, and it is crucially related to the statistical independence of scatterers in a complex scene. Although this would seem to allow PolInSAR to overcome the difficulties involving the controversial confusion between statistical independence and polarization as present in PolSAR, Cloude's PolInSAR originally inherited the idea of separating physical contributors to the scattering phenomenon through the use of singular values and vectors. This was an assumption consistent with Cloude's PolSAR postulates that was later set aside. We propose the introduction of a multidimensional coherence tensor that includes PolInSAR's polarimetric interferometry matrix Ω12 as its 2-D case. We show that some important properties of the polarimetric interferometry matrix are incidental to its bidimensionality. Notably, this exceptional behavior in 2-D seems to suggest that the singular value decomposition (SVD) of Ω12 does not provide a physical insight into the scattering problem in the sense of splitting different scattering contributors. It might be argued that Cloude's PolInSAR in its current form does not rely on the SVD of Ω12 but on other underlying optimization schemes. The drawbacks of such ulterior developments and the failure of the maximum coherence separation procedure to be a consistent scheme for surface topography estimation in a two-layer model are discussed in depth in this paper. Nevertheless, turning back to the SVD of Ω12, the use of the singular values of a prewhitened version of Ω12 is consistent with a leading method of characterizing coherence in modern Optics. For this reason, the utility of the SVD of Ω12 as a means of characterizing coherence is analyzed here and extended to higher dimensionalities. Finally, these extensions of the concept of coherence to the multidimensional case are tested and compared with the 2-D case by numerically simulating the scattered electromagnetic field from a rough surface. [ABSTRACT FROM PUBLISHER]