We use the Laplace/Borel sum rules (LSR) and the finite energy/local duality sum rules (FESR) to investigate the non-strange $udbar ubar d$ and hidden-strange $usbar ubar s$ tetraquark states with exotic quantum numbers $J^{PC}=0^{+-}$ . We systematically construct all eight possible tetraquark currents in this channel without covariant derivative operator. Our analyses show that the $udbar ubar d$ systems have good behaviour of sum rule stability and expansion series convergence in both the LSR and FESR analyses, while the LSR for the $usbar ubar s$ states do not associate with convergent OPE series in the stability regions and only the FESR can provide valid results. We give the mass predictions $1.43pm0.09$ GeV and $1.54pm0.12$ GeV for the $udbar ubar d$ and $usbar ubar s$ tetraquark states, respectively. Our results indicate that the $0^{+-}$ isovector $usbar ubar s$ tetraquark may only decay via weak interaction mechanism, e.g. $X_{usbar{u}bar{s}}to Kpipi$, since its strong decays are forbidden by kinematics and the symmetry constraints on the exotic quantum numbers. It is predicted to be very narrow, if it does exist. The $0^{+-}$ isoscalar $usbar ubar s$ tetraquark is also predicted to be not very wide because its dominate decay mode $X_{usbar{u}bar{s}}tophipipi$ is in $P$-wave.
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