We compare two rotationally invariant decomposition techniques on linear polarisation data: the spin-2 spherical harmonic decomposition in two opposite parities, the $E$- and $B$-mode, and the multiscale analysis of the gradient of linear polarisation, $| abla mathbf{P}|$. We demonstrate that both decompositions have similar properties in the image domain and the spatial frequency domain. They can be used as complementary tools for turbulence analysis of interstellar magnetic fields in order to develop a better understanding of the origin of energy sources for the turbulence, the origin of peculiar magnetic field structures and their underlying physics. We also introduce a new quantity $| abla EB|$ based on the $E$- and $B$-modes and we show that in the intermediate and small scales limit $| abla EB| simeq | abla mathbf{P}|$. Analysis of the 2.3 GHz S-band Polarization All Sky Survey (S -PASS) shows many extended coherent filament-like features appearing as double-jumps in the $| abla mathbf{P}|$ map that are correlated with negative and positive filaments of $B$-type polarisation. These local asymmetries between the two polarisation types, $E$ and $B$, of the non-thermal Galactic synchrotron emission have an influence on the $E$- and $B$-mode power spectra analyses. The wavelet-based formalism of the polarisation gradient analysis allows us to locate the position of $E$- or $B$-mode features responsible for the local asymmetries between the two polarisation types. In analysed subregions, the perturbations of the magnetic field are trigged by star clusters associated with HII regions, the Orion-Eridanus superbubble and the North Polar Spur at low Galactic latitude.
Download