Reservoir computing has become immensely popular for exploiting physical systems for computation. A multitude of physical reservoirs have been demonstrated, ranging from assembles of nanomagnets to living cultures of neurons. Unlike abstract software reservoirs, physical reservoirs are subject to spatial constraints which restricts the possible reservoir topologies. Here, we investigate lattice reservoirs, where nodes are placed on a regular lattice which defines the reservoir topology and its weights. Despite their simple regular structure, lattice reservoirs perform surprisingly well, in some cases outcompeting classical Echo State Networks. A key finding is the need for directed edges to facilitate information flow within the reservoir, highlighting the importance of symmetry breaking in physical reservoirs. We take advantage of the spatial nature of lattice reservoirs to discover key computational structures within, revealing what these reservoirs are actually doing. Lattice reservoirs bridge the gap between physics and computation, providing invaluable insight for the design and understanding of physical reservoirs.