  We derive new bounds on Lorentz violations in the electron sector from
existing data on high-energy astrophysical sources. Synchrotron and inverse
Compton data give precisely complementary constraints. The best bound on a
specific combination of electron Lorentz-violating coefficients is at the 6 x
10^(-20) level, and independent bounds are available for all the
Lorentz-violating c coefficients at the 2 x 10^(-14) level or better. This
represents an improvement in some bounds by fourteen orders of magnitude.
