  We consider SU(2)_L x U(1) gauge invariant generalizations of a nonlocal,
Lorentz violating mass term for neutrinos that preserves a SIM(2) subgroup.
This induces Lorentz violating effects in QED as well as tree-level lepton
family number violating interactions. Measurements of g_e - 2 with trapped
electrons severely constrain possible SIM(2) mass terms for electrons which
violate C invariance. We study Lorentz violating effects in a C invariant and
SIM(2) invariant extension of QED. We examine the Lorentz violating
interactions of nonrelativistic electrons with electromagnetic fields to
determine their impact on the spectroscopy of hydrogen-like atoms and g_e - 2
measurements with trapped electrons. Generically, Lorentz violating corrections
are suppressed by m_\nu^2/m_e^2 and are within experimental limits. We study
one-loop corrections to electron and photon self-energies and point out the
need for a prescription to handle IR divergences induced by the nonlocality of
the theory. We also calculate the tree level contribution to \mu to e + \gamma
from SIM(2) invariant mass terms.
