  We find an interesting fact that fictitious retrograde co-orbitals of Saturn,
or small bodies inside the retrograde 1:1 resonance with Saturn, are highly
unstable in our numerical simulations. It is shown that in the presence of
Jupiter, the retrograde co-orbitals will get ejected from Saturn's co-orbital
space within a timescale of 10 Myr. This scenario reminds us of the instability
of Saturn Trojans caused by both the Great Inequality and the secular
resonances. Therefore, we carry out in-depth inspections on both mechanisms and
prove that the retrograde resonance overlap, raised by Great Inequality, cannot
serve as an explanation for the instability of retrograde co-orbitals, due to
the weakness of the retrograde 2:5 resonance with Jupiter at a low
eccentricity. However, we discover that both $\nu_5$ and $\nu_6$ secular
resonances contribute to the slow growth of the eccentricity, therefore, are
possibly the primary causes of the instability inside Saturn's retrograde
co-orbital space.
