The use of meteorological radars to study snowfall microphysical properties and processes is well established, in particular via a few distinct techniques: the use of radar polarimetry, of multi-frequency radar measurements, and of the radar Doppler spectra. We propose a novel approach to retrieve snowfall properties by combining the latter two techniques, while relaxing some assumptions on, e.g., beam alignment and non-turbulent atmosphere. The method relies on a two-step deep-learning framework inspired from data compression techniques: an encoder model maps a high-dimensional signal to a low-dimensional latent space, while the decoder reconstructs the original signal from this latent space. Here, Doppler spectrograms at two frequencies constitute the high-dimensional input, while the latent features are constrained to represent the snowfall properties of interest. The decoder network is first trained to emulate Doppler spectra from a set of microphysical variables, using simulations from the Passive and Active Microwave radiative TRAnsfer model (PAMTRA) as training data. In a second step, the encoder network learns the inverse mapping, from real measured dual-frequency spectrograms to the microphysical latent space; in doing so, it leverages with a convolutional structure the spatial consistency of the measurements to mitigate the ill-posedness of the problem. The method was implemented on X-and W-band data from the ICE GENESIS campaign that took place in the Swiss Jura Mountains in January 2021. An in-depth assessment of the retrieval accuracy was performed through comparisons with colocated aircraft in situ measurements collected during three precipitation events. The agreement is overall good and opens up possibilities for acute characterization of snowfall microphysics on larger datasets. A discussion of the sensitivity and limitations of the method is also conducted. The main contribution of this work is, on the one hand, the theoretical framework itself, which can be applied to other remote-sensing retrieval applications and is thus possibly of interest to a broad audience across atmospheric sciences. On the other hand, the seven retrieved microphysical descriptors provide relevant insights into snowfall processes.