Buildings that are rotated with respect to the sensor trajectory could be erroneously classified as vegetated areas in the Pauli basis, and subsequently in many decomposition theorems despite the considerable amount of work done to solve that issue. This misjudgement is linked to the high level of their cross-polarized contribution. Using electromagnetic simulation tools and image analysis, we study the value of these cross-polarization components. We show that forested areas and cities exhibit significantly different cross-polarization levels; indeed, the origin of these components is actually distinct. Based on that, to discriminate between the two environments, we introduce an extension to the Pauli basis where the cross polarization is split into two classes, one for rotated dihedrals and the other for random scatterers. This approach is then tested on two synthetic aperture radar images: the first acquired at C-band using RADARSAT-2 over Downtown San Francisco and the second using RAMSES at X-band over an industrial area near Paris.