  The spin symmetry in the Dirac negative energy spectrum and its origin are
investigated for the first time within the density-dependent relativistic
Hartree-Fock (DDRHF) theory. Taking the nucleus $^{16}$O as an example, the
spin symmetry in the negative energy spectrum is found to be a good
approximation and the dominant components of the Dirac wave functions for the
spin doublets are nearly identical. In comparison with the relativistic Hartree
approximation where the origin of spin symmetry lies in the equality of the
scalar and vector potentials, in DDRHF the cancellation between the Hartree and
Fock terms is responsible for the better spin symmetry properties and
determines the subtle spin-orbit splitting. These conclusions hold even in the
case when significant deviations from the G-parity values of the
meson-antinucleon couplings occur.
