©2018 American Mathematical Society. We study the closure of the cubic Principal Hyperbolic Domain and its intersection Pλ with the slice Fλ of the space of all cubic polynomials with fixed point 0 defined by the multiplier λ at 0. We show that any bounded domain W of Fλ \Pλ consists of J-stable polynomials f with connected Julia sets J(f) and is either of Siegel capture type (then f ∈ W has an invariant Siegel domain U around 0 and another Fatou domain V such that f|V is two-to-one and fk (V ) = U for some k > 0) or of queer type (then a specially chosen critical point of f ∈ W belongs to J(f), the set J(f) has positive Lebesgue measure, and it carries an invariant line field).