Many causal discovery algorithms exploit conditional independence signatures in observational data, recovering a Markov equivalence class (MEC) of possible graphs consistent with the data. In case the MEC is non-trivial, additional assumptions on the data generating process can be made, and generative models can be fit to further resolve the MEC. We show that triangular monotonic increasing (TMI) maps parametrize generative models that perform conditional independence-based causal discovery by searching over permutations, that additionally are flexible enough as generative models to fit a wide class of causal models. In this paper, we characterize the theoretical properties that make these models relevant as tools for causal discovery, make connections to existing methods, and highlight open challenges towards their deployment.