  We explore the density and spin self-ordering of driven spin-$1/2$
collisionless fermionic atoms coupled to the electromagnetic fields of a ring
resonator. The two spin states are two-photon Raman-coupled via a pair of
degenerate counterpropagating cavity modes and two transverse pump fields. In
this one-dimensional configuration the coupled atom-field system possesses a
continuous $U(1)$ translational symmetry and a discrete $\mathbf{Z}_2$ spin
inversion symmetry. At half filling for sufficiently strong pump strengths, the
combined $U(1)\times \mathbf{Z}_2$ symmetry is spontaneously broken at the
onset of a superradiant phase transition to a state with self-ordered density
and spin structures. We predominately find an antiferromagnetic lattice order
at the cavity wavelength. The self-ordered states exhibit unexpected positive
momentum pair correlations between fermions with opposite spin. These strong
cavity-mediated correlations vanish at higher pump strength.
