  At present the color-octet mechanism is still an important and debatable part
in the non-relativistic QCD(NRQCD). We find in this work that the polarized
double charmonium production at the LHC may pose a stringent test on the
charmonium production mechanism. Result shows that the transverse
momentum($p_T$) scaling behaviors of double $J/\psi$ differential cross
sections in color-singlet and -octet production mechanisms deviate
distinctively from each other while $p_T$ is larger than 7 GeV. In color-octet
mechanism, the two $J/\psi$s in one pair are mostly transversely polarized when
$p_T\gg
  2 m_c$, as expected from the fragmentation limit point of view. In
color-singlet mechanism, there is about one half of the charmonium pairs with
at least one $J/\psi$ being longitudinally polarized at moderate transverse
momentum. The energy dependence of the polarized $J/\psi$ pair production is
found to be weak, and this process is found to be experimentally attainable in
the early phase of the LHC operation.
