  We apply the QCD sum rule method to study the double-gluon hybrid states with
the quark-gluon contents $\bar q q gg$ ($q=u/d$) and $\bar s s gg$. We
construct twenty-eight double-gluon hybrid currents, eleven of which are found
to be zero due to some internal symmetries between the two gluons fields. We
concentrate on the non-vanishing currents with the exotic quantum numbers
$J^{PC} = 1^{-+}$ and $3^{-+}$. Their masses are calculated to be $M_{|\bar q q
gg;1^{-+}\rangle} = 4.35^{+0.26}_{-0.30}$ GeV, $M_{|\bar s s gg;1^{-+}\rangle}
= 4.49^{+0.25}_{-0.30}$ GeV, $M_{|\bar q q gg;3^{-+}\rangle} =
3.02^{+0.24}_{-0.31}$ GeV, and $M_{|\bar s s gg;3^{-+}\rangle} =
3.16^{+0.22}_{-0.28}$ GeV. The decay behaviors of the $J^{PC} = 3^{-+}$ states
are studied, and we propose to search for them in the $\pi a_1(1260)/\rho
\omega/\phi \phi$ channels in future particle experiments.
