  The expressive power of Graph Neural Networks (GNNs) has been studied
extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However,
standard GNNs and the WL framework are inapplicable for geometric graphs
embedded in Euclidean space, such as biomolecules, materials, and other
physical systems. In this work, we propose a geometric version of the WL test
(GWL) for discriminating geometric graphs while respecting the underlying
physical symmetries: permutations, rotation, reflection, and translation. We
use GWL to characterise the expressive power of geometric GNNs that are
invariant or equivariant to physical symmetries in terms of distinguishing
geometric graphs. GWL unpacks how key design choices influence geometric GNN
expressivity: (1) Invariant layers have limited expressivity as they cannot
distinguish one-hop identical geometric graphs; (2) Equivariant layers
distinguish a larger class of graphs by propagating geometric information
beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable
maximally powerful geometric GNNs; and (4) GWL's discrimination-based
perspective is equivalent to universal approximation. Synthetic experiments
supplementing our results are available at
\url{https://github.com/chaitjo/geometric-gnn-dojo}
