  We present a calculation of the fourth-order (NLO) contribution to
spin-dependent Compton scattering in heavy-baryon chiral perturbation theory,
and we give results for the four spin polarisabilities. No low-energy
constants, except for the anomalous magnetic moments of the nucleon, enter at
this order. For forward scattering the fourth-order piece of the spin
polarisability of the proton turns out to be almost twice the size of the
leading piece, with the opposite sign. This leads to the conclusion that no
prediction can currently be made for this quantity. For backward scattering the
fourth-order contribution is much smaller than the third-order piece which is
dominated by the anomalous scattering, and so cannot explain the discrepancy
between the CPT result and the current best experimental determination.
