  We study the possible exotic states with $J^{PC} = 0^{+-}$ using the
tetraquark interpolating currents with the QCD sum rule approach. The extracted
masses are around 4.85 GeV for the charmonium-like states and 11.25 GeV for the
bottomomium-like states. There is no working region for the light tetraquark
currents, which implies the light $0^{+-}$ state may not exist below 2 GeV.
