The hemispherical power asymmetry observed by Planck and WMAP can be interpreted as due to a spatially-varying and scale-dependent component of the adiabatic power spectrum. We derive general constraints on the magnitude and scale-dependence of a component with a dipole spatial variation. The spectral index and the running of the spectral index can be significantly shifted from their inflation model values, resulting in a smaller spectral index and a more positive running. A key prediction is a hemispherical asymmetry of the spectral index and of its running. Measurement of these asymmetries can test the structure of the perturbation responsible for the cosmic microwave background power asymmetry.