  We introduce a novel symmetry for quantum 6j-symbols, which we call the
tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any
representations, including ones with multiplicities. We provide several
evidences in favour of the tug-the-hook symmetry. First, this symmetry follows
from the eigenvalue conjecture. Second, it is shown by several new examples of
explicit coincidence of 6j-symbols with multiplicities. Third, the tug-the-hook
symmetry for Wilson loops for knots in the 3d Chern-Simons theory implies the
tug-the-hook symmetry for quantum 6j-symbols. An important implication of the
analysis is the generalization of the tug-the-hook symmetry for the
Chern-Simons Wilson loops to the case of links.
