  Deviations from relativity are tightly constrained by numerous experiments. A
class of unmeasured and potentially large violations is presented that can be
tested in the laboratory only via weak gravity couplings. Specialized highly
sensitive experiments could achieve measurements of the corresponding effects.
A single constraint of 1 x 10^{-11} GeV is extracted on one combination of the
12 possible effects in ordinary matter. Estimates are provided for attainable
sensitivities in existing and future experiments.
