  We present explicit formulas for the operators providing missing labels for
the tensor product of two irreducible representations of $\mathfrak{su}_3$. The
result is seen as a particular representation of the diagonal centraliser of
$\mathfrak{su}_3$ through a pair of tridiagonal matrices. Using these explicit
formulas, we investigate the symmetry of this missing label problem and we find
a symmetry group of order 144 larger than what can be expected from the natural
symmetries. Several realisations of this symmetry group are given, including an
interpretation as a subgroup of the Weyl group of type $E_6$, which appeared in
an earlier work as the symmetry group of the diagonal centraliser. Using the
combinatorics of the root system of type $E_6$, we provide a family of
representations of the diagonal centraliser by infinite tridiagonal matrices,
from which all the finite-dimensional representations affording the missing
label can be extracted. Besides, some connections with the Hahn algebra,
Heun--Hahn operators and Bethe ansatz are discussed along with some
similarities with the well-known symmetries of the Clebsch--Gordan
coefficients.
