  We study quantum degenerate ultra-cold Fermi gases in the presence of
artificial spin-orbit coupling and crossed Zeeman fields. We emphasize the case
where parity is violated in the excitation spectrum and compare it with the
simpler situation where parity is preserved. We investigate in detail
spectroscopic properties such as the excitation spectrum, the spectral
function, momentum distribution and density of states for the cases where
parity is preserved or violated. Similarly, we show that thermodynamic
properties such as pressure, chemical potential, entropy, specific heat,
isothermal compressibility and induced spin polarization become anisotropic as
a function of Zeeman field components, when parity is violated. Lastly, we
discuss the effects of interactions and present results for the pairing
temperature as the precursor for the transition to a superfluid state. In
particular, we find that the pairing temperature is dramatically reduced in the
weak interaction regime as parity violation gets stronger, and that the
momentum dependence of the order parameter for superfluidity violates parity
when crossed Zeeman fields are present for finite spin-orbit coupling.
