  Miquel dynamics were introduced by Ramassamy as a discrete time evolution of
square grid circle patterns on the torus. In each time step every second circle
in the pattern is replaced with a new one by employing Miquel's six circle
theorem. Inspired by these dynamics we define the Miquel move, which changes
the combinatorics and geometry of a circle pattern locally. We prove that the
circle centers under Miquel dynamics are Clifford lattices, considered as an
integrable system by Konopelchenko and Schief. Clifford lattices have the
combinatorics of an octahedral lattice and every octahedron contains six
intersection points of Clifford's four circle configuration. The Clifford move
replaces one of these circle intersection points with the opposite one. We
establish a new connection between circle patterns and the dimer model: If the
distances between circle centers are interpreted as edge weights, the Miquel
move preserves probabilities in the sense of of urban renewal.
