The decay B0 → DK*0 and the charge conjugate mode are studied using 1.0 fb−1 of pp collision data collected by the LHCb experiment at √s = 7 TeV in 2011. The CP asymmetry between the B0 → DK*0 and the B0 → DK*0 decay rates, with the neutral D meson in the CP-even final state K+K−‚ is found to be AdKK = −0.45 ± 0.23 ± 0.02, where the first uncertainty is statistical and the second is systematic. In addition, favoured B0 → DK*0 decays are reconstructed with the D meson in the non-CP eigenstate K+π−. The ratio of the B-flavour averaged decay rates in D decays to CP and non-CP eigenstates is measured to be RdKK = 1.36 +0.37−0.32 ± 0.07, where the ratio of the branching fractions of D0 → K−π+ to D0 → K+K− decays is included as multiplicative factor. The CP asymmetries measured with two control channels, the favoured B0 → DK*0 decay with D → K+π− and the B0s → DK*0 decay with D → K+K−, are also reported.