The statistical characterization of the diffuse magnetized interstellar medium (ISM) and Galactic foregrounds to the cosmic microwave background (CMB) poses a major challenge. To account for their non-Gaussian statistics, we need a data analysis approach capable of efficiently quantifying statistical couplings across scales. This information is encoded in the data, but most of it is lost when using conventional tools, such as one-point statistics and power spectra. The wavelet scattering transform (WST), a low-variance statistical descriptor of non-Gaussian processes introduced in data science, opens a path towards this goal. To establish the methodology, we applied the WST to noise-free maps of dust polarized thermal emission computed from a numerical simulation of magnetohydrodynamical turbulence in the diffuse ISM. We analyzed normalized complex Stokes maps and maps of the polarization fraction and polarization angle. The WST yields a few thousand coefficients; some of them measure the amplitude of the signal at a given scale, and the others characterize the couplings between scales and orientations. The dependence on orientation can be fitted with the reduced wavelet scattering transform (RWST), an angular model introduced in previous works for total intensity maps. The RWST provides a statistical description of the polarization maps, quantifying their multiscale properties in terms of isotropic and anisotropic contributions. It allowed us to exhibit the dependence of the map structure on the orientation of the mean magnetic field and to quantify the non-Gaussianity of the data. We also used RWST coefficients, complemented by additional constraints, to generate random synthetic maps with similar statistics. Their agreement with the original maps demonstrates the comprehensiveness of the statistical description provided by the RWST. This work is a step forward in the analysis of observational data and the modeling of CMB foregrounds. We also release PyWST, a public Python package to perform WST and RWST analyses of two-dimensional data.Key words: dust, extinction / ISM: magnetic fields / turbulence / methods: statistical / polarization★ The public Python package PyWST is available at https://github.com/bregaldo/pywst