In a previous joint experimental and theoretical study of the barrierless chemical reaction C(1D) + H2 at low temperatures (300-50 K) [K. M. Hickson, J.-C. Loison, H. Guo, Y. V. Suleimanov, J. Phys. Chem. Lett., 2015, 6, 4194.], excellent agreement was found between experimental thermal rate constants and theoretical estimates based on ring polymer molecular dynamics (RPMD) over the two lowest singlet potential energy surfaces (PESs). Here, we extend this work to one of its deuterated counterparts, C(1D) + D2, over the same temperature range. Experimental and RPMD results are in very good agreement when contributions from both PESs to this chemical reaction are included in the RPMD simulations. The deviation between experiment and the RPMD calculations does not exceed 25 % and both results exhibit a slight negative temperature dependence. The first excited 1A" PES plays a more important role than the ground 1A' PES as the temperature is decreased, similar to our previous studies of the C(1D) + H2 reaction but with a more pronounced effect. The small differences in temperature dependence between the earlier and present experimental studies of C(1D) + H2/D2 reactions are discussed in terms of the use of non-equilibrium populations of ortho/para-H2/D2. We argue that RPMD provides a very convenient and reliable tool to study low-temperature chemical reactions.