The interstellar medium (ISM) is a complex nonlinear system governed by the interplay between gravity and magneto-hydrodynamics, as well as radiative, thermodynamical, and chemical processes. Our understanding of it mostly progresses through observations and numerical simulations, and a quantitative comparison between these two approaches requires a generic and comprehensive statistical description of the emerging structures. The goal of this paper is to build such a description, with the purpose of permitting an efficient comparison that is independent of any specific prior or model. We started from the wavelet scattering transform (WST), a low-variance statistical description of non-Gaussian processes, which was developed in data science and encodes long-range interactions through a hierarchical multiscale approach based on the wavelet transform. We performed a reduction of the WST through a fit of its angular dependencies. This allowed us to gather most of the information it contains into a few components whose physical meanings are identified and describe for instance isotropic and anisotropic behaviours. The result of this paper is the reduced wavelet scattering transform (RWST), a statistical description with a small number of coefficients that characterizes complex structures arising from nonlinear phenomena, in particular interstellar magnetohydrodynamical (MHD) turbulence, independently of any specific priors. The RWST coefficients encode moments of order up to four, have reduced variances, and quantify the couplings between scales. To show the efficiency and generality of this description, we applied it successfully to the following three kinds of processes that are a priori very different: fractional Brownian motions, MHD simulations, and Herschel observations of the dust thermal continuum in a molecular cloud. With fewer than 100 RWST coefficients when probing six scales and eight angles on 256 by 256 maps, we were able to perform quantitative comparisons, infer relevant physical properties, and produce realistic synthetic fields.