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One of the great mysteries about MnWO₄ is how two different multiferroic and cycloidal phases, labeled AF2 and AF5, can be produced by Co or Zn doping. An analysis of inelastic neutron-scattering data is used to determine the underlying exchange interactions and anisotropies for the AF5 phase of 12% Co-doped MnWO₄. Although the ordering wavevector Q = (0.23, 0.5, -0.46) of the AF5 phase is incommensurate in two dimensions, a new technique reduces the computation of the spin-wave frequencies and intensities to a one-dimensional problem. Excellent fits to the inelastic spectra are obtained by including the 11 competing exchange interactions and easy-plane anisotropy without the Dzyaloshinskii-Moriya interactions that are believed to be important for the cycloidal AF2 phase. Remarkably, the exchange parameters of the AF5 phase of Co-doped MnWO₄ are relatively close to those of the collinear AF1 phase of pure MnWO₄, which has also been reconsidered in this work. So the primary effect of Co doping is to rotate the single-ion anisotropy from easy axis to easy plane.