Questioning basic assumptions about the structure of space and time has greatly enhanced our understanding of nature. State-of-the-art atomic clocks^{, –} make it possible to precisely test fundamental symmetry properties of spacetime and search for physics beyond the standard model at low energies of just a few electronvolts. Modern tests of Einstein’s theory of relativity try to measure so-far-undetected violations of Lorentz symmetry; accurately comparing the frequencies of optical clocks is a promising route to further improving such tests. Here we experimentally demonstrate agreement between two single-ion optical clocks at the 10⁻¹⁸ level, directly validating their uncertainty budgets, over a six-month comparison period. The ytterbium ions of the two clocks are confined in separate ion traps with quantization axes aligned along non-parallel directions. Hypothetical Lorentz symmetry violations^{, –} would lead to periodic modulations of the frequency offset as the Earth rotates and orbits the Sun. From the absence of such modulations at the 10⁻¹⁹ level we deduce stringent limits of the order of 10⁻²¹ on Lorentz symmetry violation parameters for electrons, improving previous limits^{, –} by two orders of magnitude. Such levels of precision will be essential for low-energy tests of future quantum gravity theories describing dynamics at the Planck scale, which are expected to predict the magnitude of residual symmetry violations.