We use the Laplace/Borel sum rules (LSR) and the finite energy/local duality sum rules (FESR) to investigate the nonstrange $ud\overline{u}\overline{d}$ and hidden-strange $us\overline{u}\overline{s}$ tetraquark states with exotic quantum numbers $J^{PC}=0^{+-}$. We systematically construct all eight possible tetraquark currents in this channel without a covariant derivative operator. Our analyses show that the $ud\overline{u}\overline{d}$ systems have good behavior of sum rule stability and expansion series convergence in both the LSR and FESR analyses, while the LSR for the $us\overline{u}\overline{s}$ 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 $1.43\pm 0.09\mathrm{GeV}$ and $1.54\pm 0.12\mathrm{GeV}$ for the $ud\overline{u}\overline{d}$ and $us\overline{u}\overline{s}$ tetraquark states, respectively. Our results indicate that the $0^{+-}$ isovector $us\overline{u}\overline{s}$ tetraquark may only decay via weak interaction mechanism, e.g., $X_{us\overline{u}\overline{s}}\to K\pi \pi$, 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 $us\overline{u}\overline{s}$ tetraquark is also predicted to be not very wide because its dominate decay mode $X_{us\overline{u}\overline{s}}\to \varphi \pi \pi$ is in $P$ wave.