We compute the spectrum of the Schreier graph of the symmetric group Sn corresponding to the Young subgroup S2×Sn−2 and the generating set consisting of initial reversals. In particular, we show that this spectrum is integral and for n≥8 consists precisely of the integers {0,1,…,n}. A consequence is that the first positive eigenvalue of the Laplacian is always 1 for this family of graphs.