  We present the leading order coupling of double-heavy hadrons to heavy hadron
pairs in Born-Oppenheimer effective field theory. We obtain the expressions for
the contribution of heavy hadron pairs to the masses and widths of double-heavy
hadrons. We apply our result for the specific case of the coupling of the
lowest lying heavy hybrids and
$D_{(s)}^{(*)}\bar{D}_{(s)}^{(*)}(B_{(s)}^{(*)}\bar{B}_{(s)}^{(*)})$ obtaining
a set of selection rules for the decays. We build a model for the coupling
potential and compute the corresponding decay widths and the contributions to
the mass of the heavy hybrids. We compare our results with the experimental
exotic quarkonium spectrum and discuss the most likely experimental candidates
for quarkonium hybrids.
