Recognizing precise geometrical configurations of groups of objects is a key capability of human spatial cognition, yet little studied in the deep learning literature so far. In particular, a fundamental problem is how a machine can learn and compare classes of geometric spatial configurations that are invariant to the point of view of an external observer. In this paper we make two key contributions. First, we propose SpatialSim (Spatial Similarity), a novel geometrical reasoning benchmark, and argue that progress on this benchmark would pave the way towards a general solution to address this challenge in the real world. This benchmark is composed of two tasks: Identification and Comparison, each one instantiated in increasing levels of difficulty. Secondly, we study how relational inductive biases exhibited by fully-connected message-passing Graph Neural Networks (MPGNNs) are useful to solve those tasks, and show their advantages over less relational baselines such as Deep Sets and unstructured models such as Multi-Layer Perceptrons. Finally, we highlight the current limits of GNNs in these tasks.