Thermo-viscoelastic linear-response functions are calculated from the master equation describing viscous liquid inherent dynamics. From the imaginary parts of the frequency-dependent isobaric specific heat, isothermal compressibility, and isobaric thermal expansion coefficient, we define a "linear dynamic Prigogine-Defay ratio" ΛT p (ω) with the property that if ΛT p (ω) = 1 at one frequency, then ΛT p (ω) is unity at all frequencies. This happens if and only if there is a single-order-parameter description of the thermo-viscoelastic linear responses via an order parameter (which may be nonexponential in time). Generalizations to other cases of thermodynamic control parameters than temperature and pressure are treated in an Appendix.