  We present a quenched calculation of lowest bottomonium hybrid states on an
anisotropic lattice with Landau mean-link tadpole improvement, using the
improved NRQCD Hamiltonian. We investigate the quark-spin s=0,1 sectors, which
contain both the non-exotic 1-- and the exotic 1-+, and demonstrate their
degeneracy in the case of the lowest order Hamiltonian. Both states are found
at around 1.6 GeV above the Upsilon ground state. We examine the spin-splitting
for several hybrid states that is due to the magnetic moment term in the NRQCD
Hamiltonian, all other terms having negligible effect. The spin splittings are
well resolved outside errors and are surprisingly large, and we investigate
their dependence on the strength of the magnetic term. We calculate one
contribution to these splittings using the Born-Oppenheimer picture for hybrids
and show that the observed size of the effect is plausibly explained by mixing
with hybrid states with more than one constituent gluon.
