Kinematic surveys of the dwarf spheroidal (dSph) satellites of the Milky Way are revealing tantalizing hints about the structure of dark matter (DM) haloes at the low-mass end of the galaxy luminosity function. At the bright end, modelling of spiral galaxies has shown that their rotation curves are consistent with the hypothesis of a universal rotation curve whose shape is supported by a cored dark matter halo. In this paper, we investigate whether the internal kinematics of the Milky Way dSphs are consistent with the particular cored DM distributions which reproduce the properties of spiral galaxies. Although the DM densities in dSphs are typically almost two orders of magnitude higher than those found in (larger) disc systems, we find consistency between dSph kinematics and Burkert DM haloes whose core radii r0 and central densities ρ0 lie on the extrapolation of the scaling law seen in spiral galaxies: log ρ0≃α log r0+ const with 0.9 < α < 1.1. We similarly find that the dSph data are consistent with the relation between ρ0 and baryon scalelength seen in spiral galaxies. While the origin of these scaling relations is unclear, the finding that a single DM halo profile is consistent with kinematic data in galaxies of widely varying size, luminosity and Hubble type is important for our understanding of observed galaxies and must be accounted for in models of galaxy formation.