  Dark radiation (DR) appears as a new physics candidate in various scenarios
beyond the Standard Model. While it is often assumed that perturbations in DR
are adiabatic, they can easily have an isocurvature component if more than one
field was present during inflation, and whose decay products did not all
thermalize with each other. By implementing the appropriate isocurvature
initial conditions (IC), we derive the constraints on both uncorrelated and
correlated DR density isocurvature perturbations from the full Planck 2018 data
alone, and also in combination with other cosmological data sets. Our study on
free-streaming DR (FDR) updates and generalizes the existing bound on neutrino
density isocurvature perturbations by including a varying number of
relativistic degrees of freedom, and for coupled DR (CDR) isocurvature, we
derive the first bound. We also show that for CDR qualitatively new physical
effects arise compared to FDR. One such effect is that for isocurvature IC, FDR
gives rise to larger CMB anisotropies compared to CDR -- contrary to the
adiabatic case. More generally, we find that a blue-tilt of DR isocurvature
spectrum is preferred. This gives rise to a larger value of the Hubble constant
$H_0$ compared to the standard $\Lambda$CDM+$\Delta N_{\rm eff}$ cosmology with
adiabatic spectra and relaxes the $H_0$ tension.
