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We study natural bases for two constructions of the irreduciblerepresentation of the symmetric group corresponding to $[n,n,n]$: the {\emreduced web} basis associated to Kuperberg's combinatorial description of thespider category; and the {\em left cell basis} for the left cell constructionof Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is theTemperley-Lieb category; reduced webs correspond to planar matchings, which areequivalent to left cell bases. This paper compares the images of these basesunder classical maps: the {\em Robinson-Schensted algorithm} betweenpermutations and Young tableaux and {\em Khovanov-Kuperberg's bijection}between Young tableaux and reduced webs.  One main result uses Vogan's generalized $\tau$-invariant to uncover a closestructural relationship between the web basis and the left cell basis.Intuitively, generalized $\tau$-invariants refine the data of the inversion setof a permutation. We define generalized $\tau$-invariants intrinsically forKazhdan-Lusztig left cell basis elements and for webs. We then show that thegeneralized $\tau$-invariant is preserved by these classical maps. Thus, ourresult allows one to interpret Khovanov-Kuperberg's bijection as an analogue ofthe Robinson-Schensted correspondence.  Despite all of this, our second main result proves that the reduced web andleft cell bases are inequivalent; that is, these bijections are not$S_{3n}$-equivariant maps.

The Robinson-Schensted Correspondence and $A_2$-web Bases