  We report on a measurement of the constancy and anisotropy of the speed of
light relative to the electrons in photon-electron scattering. We used the
Compton scattering asymmetry measured by the new Compton polarimeter in Hall~C
at Jefferson Lab to test for deviations from unity of the vacuum refractive
index ($n$). For photon energies in the range of 9 - 46 MeV, we obtain a new
limit of $1-n < 1.4 \times 10^{-8}$. In addition, the absence of sidereal
variation over the six month period of the measurement constrains any
anisotropies in the speed of light. These constitute the first study of Lorentz
invariance using Compton asymmetry. Within the minimal standard model extension
framework, our result yield limits on the photon and electron coefficients
$\tilde{\kappa}_{0^+}^{YZ}, c_{TX}, \tilde{\kappa}_{0^+}^{ZX}$, and $c_{TY}$.
Although, these limits are several orders of magnitude larger than the current
best limits, they demonstrate the feasibility of using Compton asymmetry for
tests of Lorentz invariance. Future parity violating electron scattering
experiments at Jefferson Lab will use higher energy electrons enabling better
constraints.
