Molecular clouds show complex structures reflecting their non-linear dynamics. Many studies, investigating the bridge between their morphology and physical properties, have shown the interest provided by non-Gaussian higher-order statistics to grasp physical information. Yet, as this bridge is usually characterized in the supervised world of simulations, transferring it onto observations can be hazardous, especially when the discrepancy between simulations and observations remains unknown. In this paper, we aim at identifying relevant summary statistics directly from the observation data. To do so, we develop a test that compares the informative power of two sets of summary statistics for a given dataset. Contrary to supervised approaches, this test does not require the knowledge of any data label or parameter, but focuses instead on comparing the degeneracy levels of these descriptors, relying on a notion of statistical compatibility. We apply this test to column density maps of 14 nearby molecular clouds observed by Herschel, and iteratively compare different sets of usual summary statistics. We show that a standard Gaussian description of these clouds is highly degenerate but can be substantially improved when being estimated on the logarithm of the maps. This illustrates that low-order statistics, properly used, remain a very powerful tool. We then further show that such descriptions still exhibit a small quantity of degeneracies, some of which are lifted by the higher order statistics provided by reduced wavelet scattering transforms. This property of observations quantitatively differs from state-of-the-art simulations of dense molecular cloud collapse and is not reproduced by logfBm models. Finally we show how the summary statistics identified can be cooperatively used to build a morphological distance, which is evaluated visually, and gives very satisfactory results.