  We re-consider operator mixing in the so-called $SU(2)$ sector of ${\cal N}
\, = \, 4$ super Yang-Mills theory with gauge group $SU(N)$. Where possible,
single-trace operators of moderate length are completed by higher-trace
admixtures so as to yield large $N$ tree level eigenstates.
  We are particularly interested in parity pairs with three excitations. Since
parity is respected in the mixing, the odd single-trace operators at low length
cannot receive too many admixtures. We reproduce the tree-level norms of a set
of large $N$ eigenstates up to order $1/N^4$ by integrability methods. This
involves evaluating two-point functions on the sphere, the torus, and the
double-torus. A perfect match is found as long as descendents are absent from
the mixing.
  Using twist to make the descendents appear in the integrability picture
immediately leads to the question how to modify the entangled states occurring
in the hexagon tessellations. We take a closer look at the double-trace
admixtures to the parity even three-excitation operator at length seven, which
are both products of a primary state and a descendent. Their two-point
functions are sensitive to the twist introduced into the Bethe equations. For
transverse scalar excitations we succeed in recovering the corresponding field
theory results. For longitudinal magnons our methods fail, pointing at a
potential weakness of the formalism.
