The rheological characterization of a model, nearly ideal thixotropic carbon black suspension first proposed by Dullaert and Mewis (JNNFM 139: 21-30, 2006) is extended to include large amplitude oscillatory shear (LAOS) flow, shear flow reversal, and unidirectional LAOS flow (UD-LAOS). We show how this broader data set is useful for validating and improving constitutive models of thixotropy. We apply this new data to further test a recently developed structure-kinetics model, the Modified Delaware Thixotropy Model (Armstrong et al., J Rheol. 60: 433-450, 2016a), as well as to better understand the microstructural basis and validity of strain-rate superposition methods proposed for soft solids. This comparison identifies the limitations of models based on a scalar description of microstructure, which cannot fully capture the reversal of flow directionality inherent in LAOS flow. Further, we use the model to identify a possible microstructural justification for the hypothesized technique of strain-rate frequency superposition in soft solids.