Here, I combine previously underutilized models and priors to perform more biologically-realistic phylogenetic inference from morphological data, with an example from squamate reptiles. When coding morphological characters, it is often possible to denote ordered states with explicit reference to observed or hypothetical ancestral conditions. Using this logic, we can integrate across character-state labels and estimate meaningful rates of forward and backwards transitions from plesiomorphy to apomorphy. I refer to this approach as MkA, for 'asymmetric.' The MkA model incorporates the biological reality of limited reversal for many phylogenetically informative characters, and significantly increases likelihoods in the empirical datasets. Despite this, the phylogeny of Squamata remains contentious. Total-evidence analyses using combined morphological and molecular data and the MkA approach tend towards recent consensus estimates supporting a nested Iguania. However, support for this topology is not unambiguous across datasets or analyses, and no mechanism has been proposed to explain the widespread incongruence between partitions, or the hidden support for various topologies in those partitions. Furthermore, different morphological datasets produced by different authors contain both different characters and different states for the same or similar characters, resulting in drastically different placements for many important fossil lineages. Effort is needed to standardize ontology for morphology, resolve incongruence, and estimate a robust phylogeny. The MkA approach provides a preliminary avenue for investigating morphological evolution while accounting for temporal evidence and asymmetry in character-state changes.