Quantum $6j$-symbols are the coefficients of the change of basis matrix in the central extension of Temperley-Lieb algebra(see the book by Kauffman and Lins). It is my understanding that Ocneanu has interpreted them as being coefficients of matrices of maps between the path algebras on affine $E_6$ and regular, non-affine $A_{11}$ (see this paper by Roche.)
It seems reasonable to believe that the celebrated Regge symmetry of the $6j$-symbols should correspond to some automorphism of (affine) $E_6$ (mainly because it has already been shown to correspond to some automorphism of affine $F_4$ (see this paper by Boalch.)
Is anything known about such an interpretation, or does anyone have an idea how to go about proving such a statement?