Vibrational relaxation of single-component carbon dioxide is studied using the full and reduced state-to-state models and two multi-temperature approaches. The full kinetic scheme including all vibrational states and different kinds of vibrational energy transitions within and between CO2 modes is proposed and implemented to the 0-D code for spatially homogeneous relaxation. Contributions of various energy transitions are evaluated, and dominating relaxation mechanisms are identified for two generic test cases corresponding to compression (excitation) and expansion (deactivation) regimes. It is shown that the main relaxation channels are vibrational-translation (VT) transitions in the symmetric and bending modes and two intermode vibrational-vibrational (VV) exchanges. Reduced-order models are assessed by comparisons with the results of full state-to-state simulations. The commonly used two-temperature model with the single vibrational temperature fails to describe the relaxation for all considered initial conditions. The three-temperature model provides a good agreement with the state-to-state simulations for the excitation regime, but yields a considerable discrepancy for the deactivation mode. The sources of the discrepancies are detected and several ways for the improvement of numerically efficient multi-temperature models are proposed.