  We present a calculation of the hyperfine splittings in bottomonium using
lattice Nonrelativistic QCD. The calculation includes spin-dependent
relativistic corrections through O(v^6), radiative corrections to the leading
spin-magnetic coupling and, for the first time, non-perturbative 4-quark
interactions which enter at alpha_s^2 v^3. We also include the effect of u,d,s
and c quark vacuum polarisation. Our result for the 1S hyperfine splitting is
M(Upsilon,1S) - M(eta_b,1S)= 60.0(6.4) MeV. We find the ratio of 2S to 1S
hyperfine splittings (M(Upsilon,2S) - M(eta_b,2S))/ (M(Upsilon,1S) -
M(eta_b,1S)) = 0.445(28).
