Let F be a non-Archimedean local field with finite residue field. An irreducible smooth representation of the Weil group 𝒲F of F is called essentially tame if its restriction to wild inertia is a sum of characters. Let 𝒢net (F) denote the set of isomorphism classes of irreducible, essentially tame representations of 𝒲F of dimension n. The Langlands correspondence induces a bijection of 𝒢net (F) with a certain set 𝒜net (F) of irreducible cuspidal representations of GLn(F). We work with the set Pn(F) of isomorphism classes of admissible pairs (E/F, ξ) of degree n. There is an obvious bijection of Pn(F) with 𝒢net (F) and an explicit bijection of Pn(F) with 𝒜net (F). Together, these maps give a canonical bijection of 𝒢net (F) with 𝒜net (F). We showed in an earlier paper that the Langlands correspondence is obtained by composing the map Pn (F) → 𝒜net (F) with a permutation of Pn(F) of the form (E/F, ξ) ↦ (E/F, μξξ), where μξ is a tamely ramified character of E× depending on ξ. In this paper, we show that there is a canonical choice for the character μξ and determine it explicitly.