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Lots of charmonium-like structures have been observed in the last twodecades. Most of them have quantum numbers that can be formed by a pair ofcharm and anticharm quarks, thus it is difficult to unambiguously identify theexotic ones among them. In this Letter, by exploiting heavy quark spinsymmetry, we present a robust prediction of the hadronic molecular scenario,where the $\psi(4230),\psi(4360)$ and $\psi(4415)$ are identified as $D\barD_1,D^*\bar D_1$ and $D^*\bar D^*_2$ bound states, respectively. We show that aflavor-neutral charmonium-like exotic state with quantum numbers$J^{PC}=0^{--}$, denoted as $\psi_0(4360)$, should exist as a $D^*\bar D_1$bound state. The mass and width of the $\psi_0(4360)$} are predicted to be$(4366\pm18)$ MeV and less than 10 MeV, respectively. The $\psi_0(4360)$ issignificant in two folds: no $0^{--}$ hadron has been observed so far, and astudy of this state will enlighten the understanding of the mysterious vectormesons between 4.2 and 4.5 GeV, as well as the nature of previously observedexotic $Z_c$ and $P_c$ states. We propose that such an exotic state can besearched for in $e^+e^-\to \eta \psi_0(4360)$ and uniquely identified bymeasuring the angular distribution of the outgoing $\eta$ meson.

Prediction of a narrow exotic hadronic state with quantum numbers $J^{PC}=0^{--}$