Sign languages are viso-gestual languages, using space and movement to convey meaning. To be able to transcribe them, SignWriting uses an iconic system of symbols meaningfully arranged in the page. This two-dimensional system, however, is very different to traditional writing systems, so its automatic processing poses a novel challenge for computational linguistics. In this article, we present a novel problem for the state of the art in artificial intelligence: automatic SignWriting recognition. We examine the problem, model the underlying data domain, and present a first solution in the form of an expert system that exploits the domain knowledge encoded in the data modelization. This system uses an adaptable pipeline of neural networks and deterministic processing, overcoming the challenges posed by the novelty and originality of the problem. Thanks to our data modelization, it improves the accuracy compared to a straight-forward deep learning approach by 17%. All of our data and code are publicly available, and our approach may be useful not only for SignWriting processing but also for other similar graphical data.