The Tetrahedral (or $6j$) Symbol

We will attach a scalar invariant to a tetrahedron whose edges are labelled by irreducible representations of a ternary orthogonal group ${\mathrm{S}\mathrm{O}}_3$ over a local field. This generalizes the $6j$ symbol whose theory was developed by Racah, Wigner, and Regge. We give several formulas for this invariant, including in terms of hypergeometric-type integrals and functions, and show that it admits a symmetry by the the $23040$-element Weyl group of ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{12}$. We then interpret these results in terms of relative Langlands duality, where the dual story comes from the action of ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{12}$ on a $16$-dimensional cone of spinors.