  Masses and current couplings of the charmonium and bottomonium hybrids $
\overline{c}gc$ and $\overline{b}gb$ with spin-parities $J^{\mathrm{PC}
}=0^{++},\ 0^{+-},\ 0^{-+},\ 0^{--}$ and $1^{++},\ 1^{+-},\ 1^{-+},\ 1^{--}$
are calculated using QCD two-point sum rule method. Computations are performed
by taking into account gluon condensates up to dimension 12 including terms
$\sim \langle g_{s}^{3}G^{3}\rangle ^{2}$. The parameters of the bottom-charm
hybrids $\overline{b}gc$ with quantum numbers $J^{\mathrm{PC }}=0^{+},\ 0^{-},\
1^{+}$, and $1^{-}$ are calculated as well. In computations the dominance of
the pole contribution to sum rule results is ensured. It is demonstrated that
all charmonia hybrids decay strongly to two-meson final states. The bottomonium
hybrids $0^{-+}$ and $1^{-+}$ as well as the bottom-charm hybrid mesons
$0^{-(+)}$ and $1^{-(+)}$ may be stable against strong two-meson decay modes.
Results of the present work are compared with ones obtained using the sum rule
and alternative approaches. Our predictions for parameters of the heavy hybrid
mesons may be useful to study their various decay channels which are important
for interpretation of ongoing and future experiments.
