Paper II of the Clebsch: Rigidity from Sparse Shadows series produces a signed cubic from a matching quotient. This preprint identifies its canonical five-dimensional residue as a chordal companion, distinct from the conference cubic reconstructed in Papers I and III but lying in the same two-dimensional A5-invariant pencil. The chordal cubic is singular along a rational normal quartic. Its twelve F_11-points form A5/C5, and pairing points with equal stabilizer recovers the original six-axis carrier A5/D5. The normalized outer action gives an exact oriented round trip between a selected chordal line and the conference line, with all labels, scalars, and source-return boundaries retained.