  We explore the quantum phases emerging from the interplay between spin and
motional degrees of freedom of a one-dimensional quantum fluid of spinful
fermionic atoms, effectively interacting via a photon-mediating mechanism with
tunable sign and strength g, as it can be realized in present-day experiments
with optical cavities. We find the emergence, in the very same system, of spin-
and atomic-density wave ordering, accompanied by the occurrence of
superfluidity for g > 0, while cavity photons are seen to drive strong
correlations at all g values, with fermionic character for g > 0, and bosonic
character for g < 0. Due to the long-range nature of interactions, to infer
these results we combine mean-field and exact diagonalization methods supported
by bosonization analysis.
