We use data on a cultural fitness landscape, recently inferred from a large-scale cross-cultural survey of religious practices (6000+ years, 407 cultures), to provide new insights into the dynamics of cultural macroevolution. We report three main results. First, we observe an emergent distinction between the long-run fitness of a religious practice, and its short-term stability: in particular, some low-fitness practices are nonetheless highly stable. Second, despite the exponentially large size of the landscape, we find a small number of cultural attractors, and 70% of all observed configurations flow into just four, which we label "monastic", "evangelical", "indigenous", and "pre-Axial". Finally, we find large variation in the evolvability of different traits, with some (such as a belief in punishing gods) strongly fixed by context, and others (such as belief in reincarnation) much more fluid.