A 2-local representation of the twin group reduces to an (n-1)-dimensional one, and this reduced representation is irreducible exactly when a avoids 1, -1 and roots of an explicit polynomial.
Thus, the subspace ⟨v2⟩ is invariant under the action of S2 if and only if a = 1
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On the classification and irreducibility of $2$-local representations of the twin group $T_n$
A 2-local representation of the twin group reduces to an (n-1)-dimensional one, and this reduced representation is irreducible exactly when a avoids 1, -1 and roots of an explicit polynomial.