  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 CMB power asymmetry.
