  In the presence of CP violation, the Higgs-top coupling may have both scalar
and pseudoscalar components, $\kappa_t$ and ${\tilde \kappa_t}$, which are
bounded indirectly but only weakly by the present experimental constraints on
the Higgs-gluon-gluon and Higgs-$\gamma$-$\gamma$ couplings, whereas upper
limits on electric dipole moments provide strong additional indirect
constraints on ${\tilde \kappa_t}$, if the Higgs-electron coupling is similar
to that in the Standard Model and there are no cancellations with other
contributions. We discuss methods to measure directly the scalar and
pseudoscalar Higgs-top couplings by measurements of Higgs production in
association with ${\bar t} t$, single $t$ and single ${\bar t}$ at the LHC.
Measurements of the total cross sections are very sensitive to variations in
the Higgs-top couplings that are consistent with the present indirect
constraints, as are invariant mass distributions in ${\bar t} t H$, $tH$ and
${\bar t} H$ final states. We also investigate the additional information on
$\kappa_t$ and ${\tilde \kappa_t}$ that could be obtained from measurements of
the longitudinal and transverse $t$ polarization in the different associated
production channels, and the ${\bar t} t$ spin correlation in ${\bar t} t H$
events.
