  We study the existence of fully-heavy hidden-flavor $bc\bar{b}\bar{c}$
tetraquark states with various $J^{PC}=0^{\pm+}, 0^{--},1^{\pm\pm}, 2^{++}$, by
using the moment QCD sum rule method augmented by fundamental inequalities. We
construct the diquark-antidiquark type of interpolating currents in both the
$[\mathbf{\bar{3}_c}]_{bc}\otimes[\mathbf{3_c}]_{\bar{b}\bar{c}}$ and
$[\mathbf{6_c}]_{bc}\otimes[\mathbf{\bar{6}_c}]_{\bar{b}\bar{c}}$ color
structures. Our calculation shows that the masses for the S-wave positive
parity $bc\bar{b}\bar{c}$ tetraquark states are about $12.2-12.4$ GeV in both
color configuration channels. Except for two $0^{++}$ states, such masses are
below the thresholds $T_{\eta_c\eta_b}/T_{\Upsilon\psi}$ and $T_{B_cB_c}$,
implying that these S-wave positive parity $bc\bar{b}\bar{c}$ tetraquark states
are probably stable against the strong interaction. For the P-wave negative
parity $bc\bar{b}\bar{c}$ tetraquarks, their masses in the
$[\mathbf{\bar{3}_c}]_{bc}\otimes[\mathbf{3_c}]_{\bar{b}\bar{c}}$ channel are
around $12.9-13.2$ GeV, while a bit higher in the
$[\mathbf{6_c}]_{bc}\otimes[\mathbf{\bar{6}_c}]_{\bar{b}\bar{c}}$ channel.
These negative parity $bc\bar{b}\bar{c}$ tetraquarks can decay into the $c\bar
c+b\bar b$ and $c\bar b+b\bar c$ final states via the spontaneous dissociation
mechanism, including the $J/\psi\Upsilon$, $\eta_c\Upsilon$, $J/\psi\eta_b$,
$B_c^+B_c^-$ channels.
