Topological insulators are characterized by the presence of spin-momentum-locked surface states with Dirac points that span the fundamental bulk band gap. We show by first-principles calculations that the surface state of the insulator Bi₂Te₃ survives upon moderate Mn doping of the surface layers. The spin texture of both undoped and Mn-doped Bi₂Te₃ is much more complicated than commonly believed, showing layer-dependent spin reversal and spin vortices.