Classical complex dynamics began with an interest in the topology and combinatorics of the moduli space of quadratic polynomials, notable also for its special subset, the Mandelbrot set. The moduli space of all quadratic rational maps is isomorphic to C^2, and we can also understand the space of quadratic polynomials as a special curve within this space in which one critical point is marked as a fixed point. Other natural curves within the moduli space of quadratic rational maps of longstanding interest are the curves in which one critical point is marked as in an n-cycle. In this talk, we speak about the topology and combinatorics of these curves and their bifurcation locus, paying particular attention to how structure from the Mandelbrot set ("matings" and "captures") can shed light on questions about the irreducibility of these curves.