Maxwellian ratchets are autonomous, finite-state thermodynamic engines that implement input-output informational transformations. Previous studies of these "demons" focused on how they exploit environmental resources to generate work: They randomize ordered inputs, leveraging increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir while respecting both Liouvillian state-space dynamics and the Second Law. However, to date, correctly determining such functional thermodynamic operating regimes was restricted to a very few engines for which correlations among their information-bearing degrees of freedom could be calculated exactly and in closed form---a highly restricted set. Additionally, a key second dimension of ratchet behavior was largely ignored---ratchets do not merely change the randomness of environmental inputs, their operation constructs and deconstructs patterns. To address both dimensions, we adapt recent results from dynamical-systems and ergodic theories that efficiently and accurately calculate the entropy rates and the rate of statistical complexity divergence of general hidden Markov processes. In concert with the Information Processing Second Law, these methods accurately determine thermodynamic operating regimes for finite-state Maxwellian demons with arbitrary numbers of states and transitions. In addition, they facilitate analyzing structure versus randomness trade-offs that a given engine makes. The result is a greatly enhanced perspective on the information processing capabilities of information engines. As an application, we give a thorough-going analysis of the Mandal-Jarzynski ratchet, demonstrating that it has an uncountably-infinite effective state space.