  We use the Laplace/Borel sum rules (LSR) and the finite energy/local duality
sum rules (FESR) to investigate the non-strange $ud\bar u\bar d$ and
hidden-strange $us\bar u\bar s$ 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 $ud\bar u\bar d$ systems have good behaviour of sum rule
stability and expansion series convergence in both the LSR and FESR analyses,
while the LSR for the $us\bar u\bar s$ 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 $1.43\pm0.09$ GeV and $1.54\pm0.12$ GeV
for the $ud\bar u\bar d$ and $us\bar u\bar s$ tetraquark states, respectively.
Our results indicate that the $0^{+-}$ isovector $us\bar u\bar s$ tetraquark
may only decay via weak interaction mechanism, e.g. $X_{us\bar{u}\bar{s}}\to
K\pi\pi$, 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 $us\bar u\bar s$ tetraquark is also
predicted to be not very wide because its dominate decay mode
$X_{us\bar{u}\bar{s}}\to\phi\pi\pi$ is in $P$-wave.
