  The conjoined production at the LHC of single top and Higgs boson via
$t$-channel weak boson exchange is ideal to probe the top-quark Yukawa
coupling, due to a delicate cancellation between the amplitudes with the $htt$
and the $hWW$ couplings. We find that the top quark is produced with $100\%$
polarization in the leading order, and its quantum state is determined by the
spin-vector direction in the $t$-quark rest frame. We relate the spin direction
to the four-momenta of the top, Higgs and a jet in the helicity amplitude
framework. We identify a polarization asymmetry that is sensitive to CP
violation, even after partial integration over the forward jet momentum. This
CP violating asymmetry may be observed at the LHC via the component of the
top-quark polarization that is perpendicular to the $th$ scattering plane.
