  In the framework of the Born-Oppenheimer Effective Field Theory, the
hyperfine structure of heavy quarkonium hybrids at leading order in the 1/m Q
expansion is determined by two potentials. We estimate those potentials by
interpolating between the known short distance behavior and the long distance
behavior calculated in the QCD Effective String Theory. The long distance
behavior depends, at leading order, on two parameters which can be obtained
from the long distance behavior of the heavy quarkonium potentials (up to sign
ambiguities). The short distance behavior depends, at leading order, on two
extra paramentes, which are obtained from a lattice calculation of the lower
lying charmonium hybrid multiplets. This allows us to predict the hyperfine
splitting both of bottomonium hybrids and of higher multiplets of charmonium
hybrids. We carry out a careful error analysis and compare with other
approaches.
