CMB dipole - fact or fiction?

The cosmic microwave background has a dipolar structure which is interpreted in terms of the Milky Way moving relative to the CMB, inducing Doppler-shifts in the photon energies and boosting the flux. Would it be possible to distinguish between a motion dipole and a cosmological dipole on the last-scattering surface? Why is the Doppler-boosted photon spectrum still a Planck-spectrum with the correct Wien- and Stefan-Boltzmann-laws?

bonus question: What’s \(\int\mathrm{d}x\:x\ln(x)\)?