We define
G
G
, a substructure of
E
Π
\mathcal {E}_\Pi
(the lattice of
Π
1
0
\Pi ^0_1
classes), and show that a quotient structure of
G
G
,
G
♢
G^\diamondsuit
, is isomorphic to
E
∗
\mathcal {E}^*
. The result builds on the
Δ
3
0
\Delta ^0_3
isomorphism machinery, and allows us to transfer invariant classes from
E
∗
\mathcal {E}^*
to
E
Π
\mathcal {E}_\Pi
, though not, in general, orbits. Further properties of
G
♢
G^\diamondsuit
and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.