  Analytical solutions with effective CPT-violating spin--gravity corrections
to the neutron's gravitational bound states are obtained. The
helicity-dependent phase evolution due to $\vec{\sigma}\cdot\vec{\tilde{b}}$
and $\vec{\sigma}\cdot\hat{\vec{p}}$ couplings not only leads to spin
precession, %shown by the tip of the spin on the Bloch sphere,but also to
transition-frequency shifts between different gravitational bound states.
Utilizing transition frequencies measured in the qBounce experiment,\cite{GRS}
we obtain the rough bound $|\vec{\tilde{b}}|<6.9\times10^{-21}$GeV.
Incorporating known systematic errors may lead to more robust and tighter
constraints.
