  We describe the connection between inversion symmetry breaking and
criticality in free fermionic lattice models. It is shown that for
translation-invariant spinless fermions, the breaking of this symmetry in the
ground state implies criticality, i.e., the existence of long-range
correlations and the vanishing of the spectral gap; while for models with spin,
only the asymmetry of the spin-averaged covariance matrix implies a similar
conclusion. Our results are proved by introducing invariants under global
translation-invariant free fermion quenches. Using this result, we identify a
set of models where the generalized Hartree-Fock approximation must break down.
