Abstract The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called \emph{webs} and the subset of \emph{reduced webs} forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensor powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the $3n$th tensor power of $V^+$ correspond bijectively with $[n,n,n]$ standard Young tableaux. Khovanov--Kuperberg (1999) originally defined this map in terms of a graphical algorithm, while Tymoczko (2012) introduced an efficient algorithm for computing the inverse. The main result of this paper is a redefinition of Khovanov--Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan--Lusztig theory to show that Khovanov--Kuperberg's map is a direct analogue of the Robinson--Schensted correspondence. This is joint work with Heather Russell and Julianna Tymoczko.