Here we demonstrate that a simple thixotropic constitutive model produces unique signatures in large-amplitude oscillatory shear (LAOS) distinct from other nonlinear mechanisms and separate from viscoelastic time dependence. Our approach is to define the simplest model that produces the essential features of both thixotropy and viscoelasticity, a structure-parameter evolution equation coupled to a three-element fluid (Jeffreys model). In strain-controlled LAOS, the response of the model depends on four dimensionless parameters: two deformation parameters (Deborah and Weissenberg) and two material parameters (the ratio of viscoelastic to thixotropic timescales and the ratio of infinite shear viscosity to aggregate viscosity). We present numerical results for the full nonlinearities across this four-dimensional parameter space. The dimensionality is reduced by considering the asymptotically-nonlinear regime (Weissenberg number expansion). We present the first analytical solution for a thixotropic model in this asymptotically-nonlinear LAOS regime, which produces distinct power function scaling not predicted by other known solutions to nonlinear viscoelastic models. With this separation of thixotropic from viscoelastic timescales, this canonical model predicts that short thixotropic timescales can be experimentally observed with nonlinear oscillatory deformation. This is relevant to recent suggestions in distinguishing thixotropic versus "simple" yield stress fluids with no experimentally observable thixotropy. (C) 2014 Elsevier B.V. All rights reserved.