The thermal and flow characteristics of nonequilibrium monatomic, diatomic, and polyatomic gases in cylindrical Couette flow are investigated using first- and second-order Boltzmann-Curtiss-based constitutive models. The mixed modal discontinuous Galerkin scheme is used for solving the conservation laws in conjunction with the Maxwell velocity-slip and Smoluchowski temperature-jump boundary conditions. Also derived are new analytic solutions for compressible cylindrical Couette gas flow including the temperature profile, and they are used to verify the numerical scheme. Further, the second-order non-Navier-Fourier constitutive relations are derived for the cylindrical coordinates. Various abnormal behaviour is found in the second-order constitutive model, such as non-zero normal stress and excess normal stress, non-zero tangential heat flux, and flattened pressure and density profiles. The physical mechanisms behind this abnormal behaviour are found to be similar to the Knudsen layer in planar Couette gas flow, and the curvature of the cylindrical geometry does not affect the fundamental second-order physics. Moreover, two new abnormal mechanisms are found in diatomic and polyatomic gases: (i) the subtle interplay of excess normal stress (and bulk viscosity) with the nonlinear coupled constitutive relation, and (ii) the combined role of the bulk viscosity ratio and the specific heat ratio.