A topological insulator is realized via band inversions driven by the spin-orbit interaction. In the case of Z₂ topological phases, the number of band inversions is odd and time-reversal invariance is a further unalterable ingredient. For topological crystalline insulators, the number of band inversions may be even but mirror symmetry is required. Here, we prove that the chalcogenide Bi₂Te₃ is a dual topological insulator: it is simultaneously in a Z₂ topological phase with Z₂ invariants (v₀;v₁v₂v₃) = (1;000) and in a topological crystalline phase with mirror Chern number -1. In our theoretical investigation we show in addition that the Z₂2 phase can be broken by magnetism while keeping the topological crystalline phase. As a consequence, the Dirac state at the (111) surface is shifted off the time-reversal invariant momentum Γ; being protected by mirror symmetry, there is no band gap opening. Our observations provide theoretical groundwork for opening the research on magnetic control of topological phases in quantum devices.