Clock-comparison experiments are among the sharpest existing tests of Lorentz symmetry in matter. We characterize signals in these experiments arising from modifications to electron or nucleon propagators and involving Lorentz- and C P T -violating operators of arbitrary mass dimension. The spectral frequencies of the atoms or ions used as clocks exhibit perturbative shifts that can depend on the constituent-particle properties and can display sidereal and annual variations in time. Adopting an independent-particle model for the electronic structure and the Schmidt model for the nucleus, we determine observables for a variety of clock-comparison experiments involving fountain clocks, comagnetometers, ion traps, lattice clocks, entangled states, and antimatter. The treatment demonstrates the complementarity of sensitivities to Lorentz and C P T violation among these different experimental techniques. It also permits the interpretation of some prior results in terms of bounds on nonminimal coefficients for Lorentz violation, including first constraints on nonminimal coefficients in the neutron sector. Estimates of attainable sensitivities in future analyses are provided. Two technical appendices collect relationships between spherical and Cartesian coefficients for Lorentz violation and provide explicit transformations converting Cartesian coefficients in a laboratory frame to the canonical Sun-centered frame.