  We investigate positive-parity states of $^{10}$B with the calculation of
antisymmetrized molecular dynamics focusing on $pn$ pair correlations. We
discuss effects of the spin-orbit interaction on energy spectra and $pn$
correlations of the $J^\pi T=1^+_10$, $=3^+_10$, and $0^+_11$ states. The
$1^+_10$ state has almost no energy gain of the spin-orbit interaction, whereas
the $3^+_10$ state gains the spin-orbit interaction energy largely to come down
to the ground state. We interpret a part of the two-body spin-orbit interaction
in the adopted effective interactions as a contribution of the genuine $NNN$
force, and find it to be essential for the level ordering of the $3^+_10$ and
$1^+_10$ states in $^{10}$B. We also apply a $2\alpha+pn$ model to discuss
effects of the spin-orbit interaction on $T=0$ and $T=1$ $pn$ pairs around the
2$\alpha$ core. In the spin-aligned $J^\pi T=3^+0$ state, the spin-orbit
interaction affects the $(ST)=(10)$ pair attractively and keeps the pair close
to the core, whereas, in the $1^+0$ state, it gives a minor effect to the
$(ST)=(10)$ pair. In the $0^+1$ state, the $(ST)=(01)$ pair is somewhat
dissociated by the spin-orbit interaction.
