  Questioning the presumably most basic assumptions about the structure of
space and time has revolutionized 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 energy scales of just a few electron volts. Here, we experimentally
demonstrate for the first time agreement of two single-ion clocks at the
$10^{-18}$ level and directly confirm the validity of their uncertainty budgets
over a half-year long comparison period. The two clock ions are confined in
separate ion traps with quantization axes aligned along nonparallel directions.
Hypothetical Lorentz symmetry violations would lead to sidereal modulations of
the frequency offset. From the absence of such modulations at the $10^{-19}$
level we deduce stringent limits on Lorentz symmetry violation parameters for
electrons in the range of $10^{-21}$, improving previous limits by two orders
of magnitude.
