  A new sum rule for $\gamma_\pi$, the backward spin polarizability of the
nucleon, is derived from a backward-angle dispersion relation. Taking into
account single- and multi-pion photoproduction in the s-channel up to the
energy 1.5 GeV and resonances in the t-channel with mass below 1.5 GeV, it is
found for the proton and neutron that $[\gamma_\pi]_p$ = -39.5 +/- 2.4 and
$[\gamma_\pi]_n$ = 52.5 +/- 2.4, respectively, in units of 10^{-4} fm^4.
