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Dark radiation (DR) is generally predicted in new physics scenarios that address fundamental puzzles of the Standard Model or tensions in the cosmological data. Cosmological data has the sensitivity to constrain not only the energy density of DR, but also whether it is interacting. In this paper, we present a systematic study of five types of interacting DR (free-streaming, fluid, decoupling, instantaneous decoupling, and recoupling DR) and their impact on cosmological observables. We modify the Boltzmann hierarchy to describe all these types of interacting DR under the relaxation time approximation. We, for the first time, robustly calculate the collision terms for recoupling scalar DR and provide a better estimation of the recoupling transition redshift. We demonstrate the distinct features of each type of DR on the CMB and matter power spectra. We perform MCMC scans using the Planck 2018 data and BAO data. Assuming no new physics in the SM neutrino sector, we find no statistically significant constraints on the couplings of DR, although there is a slight preference for the fluid-like limit of all the cases. In the case of instantaneous decoupling DR, this limit corresponds to a late transition redshift around recombination. The $ΔN_{\rm eff}$ constraint varies marginally depending on the type of DR.