  The Jump Inversion Theorem says that for every real $A \ge_T 0'$ there is a
real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0'$. A known refinement of
this theorem says that we can choose $B$ to be a member of any special
$\Pi^0_1$ subclass of $\{0,1\}^\omega$. We now consider the possibility of
analogous refinements of two other well-known theorems: the Join Theorem -- for
all reals $A$ and $Z$ such that $A \ge_T Z \oplus 0'$ and $Z >_T 0$, there is a
real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0' \equiv_T B \oplus Z$ --
and the Pseudojump Inversion Theorem -- for all reals $A \ge_T 0'$ and every $e
\in \omega$, there is a real $B$ such that $A \equiv_T B \oplus W_e^B \equiv_T
B \oplus 0'$. We show that in these theorems, $B$ can be found in some special
$\Pi^0_1$ subclasses of $\{0,1\}^\omega$ but not in others.
