Automorphisms of the truth-table degrees are fixed on some cone

Let Dtt denote the set of truth-table degrees. A bijection π : Dtt → Dtt is an automorphism if for all truth-table degrees x and y we have x ≤tt y ⇔ π(x) ≤tt π(y). We say an automorphism π is fixed on some cone if there is a degree b such that for all x ≥tt b we have π(x) = x. We first prove that for every 2-generic real X we have X ′ 6≤tt X ⊕ 0 ′. We next prove that for every real X ≥tt 0 ′ there is a real Y such that Y ⊕ 0′ ≡tt Y ′ ≡tt X. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on some cone.