  Lots of charmonium-like structures have been observed in the last two
decades. Most of them have quantum numbers that can be formed by a pair of
charm and anticharm quarks, thus it is difficult to unambiguously identify the
exotic ones among them. In this Letter, by exploiting heavy quark spin
symmetry, we present a robust prediction of the hadronic molecular scenario,
where the $\psi(4230),\psi(4360)$ and $\psi(4415)$ are identified as $D\bar
D_1,D^*\bar D_1$ and $D^*\bar D^*_2$ bound states, respectively. We show that a
flavor-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 $\psi_0(4360)$ is
significant in two folds: no $0^{--}$ hadron has been observed so far, and a
study of this state will enlighten the understanding of the mysterious vector
mesons between 4.2 and 4.5 GeV, as well as the nature of previously observed
exotic $Z_c$ and $P_c$ states. We propose that such an exotic state can be
searched for in $e^+e^-\to \eta \psi_0(4360)$ and uniquely identified by
measuring the angular distribution of the outgoing $\eta$ meson.
