  The production of $J/\psi$-pairs as a possible measure of the polarized gluon
distribution $\Delta G(x)$ is studied for proton--nucleon collisions at
$\sqrt{s} =40\;\mbox{GeV}^2$ (HERA-$\vec{{\rm N}}$). Possibilities of
reconstructing the helicity state of at least one of the $J/\psi$'s are
critically reviewed. The observation of production asymmetries in the single
polarized mode of HERA-$\vec{{\rm N}}$ is found to be not feasible.
