The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2)-symmetry is investigated for a range of parameters in which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is bounded. By using symmetry-adapted variables, the dimension of the phase space is reduced to four. In the reduced phase space, periodic solutions are revealed as fixed points and quasiperiodic solutions as periodic orbits. Two parameter regimes are identified: A region in which trajectories in the reduced phase space are attracted by fixed points and periodic orbits and bistability occurs, and a region with complex dynamics. The characteristics of the complex dynamics are chaos, intermittency and periodic windows terminating in period doubling cascades, and both periodic and chaotic attractors occur in symmetric and asymmetric forms. The phase space geometry is suggestive of a kind of in–out intermittency involving two invariant subspaces. By using an appropriate cross-section, a reduced description of the dynamics is given in terms of a three-dimensional Poincaré map. Plots of the iterates of this map indicate that they reside in a one-dimensional manifold. In the initial range of the region with complex dynamics, the bifurcation diagrams (Poincaré map iterates versus a parameter) show the typical characteristics of unimodal map families. The symmetry of attractors is diagnosed using normalized asymmetry measures calculated from the iterates of the Poincaré map.