Recent work has shown that many cellular automata (CA) have configurations whose orbit closures are isomorphic to odometers. We investigate the geometry of the spacetime diagrams of these `odometer configurations'. For boolean linear CA, we exactly determine the positions of the consecutive `gears' of the odometer mechanism in the configuration. Then we characterize and explain the self-similar structure visible in the spacetime diagrams of odometer configurations for two classes of nonlinear CA: ratchet CA and Coven CA.