Learning faithful graph representations has become a fundamental intermediary step in a wide range of machine learning applications. We propose the systematic use of symmetric spaces as embedding targets. We use Finsler metrics integrated in a Riemannian optimization scheme, that better adapt to dissimilar structures in the graph and develop a tool to analyze the embeddings based on the vector valued distance function in a symmetric space. For implementation, we choose Siegel spaces. We show that our approach outperforms competitive baselines for graph reconstruction tasks on various synthetic and real-world datasets and further demonstrate its applicability on two downstream tasks, recommender systems and node classification. This is joint work with Federico Lopez, Beatrice Pozzetti, Michael Strube and Steve Trettel.