  Doubly heavy baryons $\left(QQq\right)$ and singly heavy antimesons
$\left(\bar{Q}q\right)$ are related by the heavy quark-diquark (HQDQ) symmetry
because in the $m_Q \to \infty$ limit, the light degrees of freedom in both the
hadrons are expected to be in identical configurations. Hyperfine splittings of
the ground states in both systems are nonvanishing at $O(1/m_Q)$ in the heavy
quark mass expansion and HQDQ symmetry relates the hyperfine splittings in the
two sectors. In this paper, working within the framework of Non-Relativistic
QCD (NRQCD), we point out the existence of an operator that couples four heavy
quark fields to the chromomagnetic field with a coefficient that is enhanced by
a factor from Coulomb exchange. This operator gives a correction to doubly
heavy baryon hyperfine splittings that scales as $1/m_Q^2 \times \alpha_S/r$,
where $r$ is the separation between the heavy quarks in the diquark. This
correction can be calculated analytically in the extreme heavy quark limit in
which the potential between the quarks in the diquark is Coulombic. In this
limit, the correction is $O(\alpha_s^2/m_Q)$ and comes with a small
coefficient. For values of $\alpha_s$ relevant to doubly charm and doubly
bottom systems, the correction to the hyperfine splittings in doubly heavy
baryons is only a few percent or smaller. We also argue that nonperturbative
corrections to the prediction for the hyperfine splittings are suppressed by
$\Lambda^2_{\rm QCD}/m_Q^2$ rather than $\Lambda_{\rm QCD}/m_Q$. Corrections
should be $\approx 10\%$ in the charm sector and smaller in heavier systems.
