On the left is the connectedness locus in the space of critically marked cubic polynomials with a fixed Siegel disk of the golden mean rotation number. There is an involution of this plane sending red to blue, which corresponds to swapping the two marked critical points. The common boundary (shown on the right) is a highly intricate Jordan curve invariant under this involution. It can be described as the locus of all cubics with both critical points on the boundary of their Siegel disks. For details, see
S. Zakeri, Dynamics of cubic Siegel polynomials, Commun. Math. Phys. 206 (1999) 185-233.