An analysis of B0→DK∗0 decays is presented, where D represents an admixture of D0 and D0 mesons reconstructed in four separate final states: K−π+, π−K+, K+K− and π+π−. The data sample corresponds to 3.0fb−1 of proton-proton collision, collected by the LHCb experiment. Measurements of several observables are performed, including CP asymmetries. The most precise determination is presented of rB(DK∗0), the magnitude of the ratio of the amplitudes of the decay B0→DK+π− with a b→u or a b→c transition, in a Kπ mass region of ±50MeV/c2 around the K∗(892) mass and for an absolute value of the cosine of the K∗0 helicity angle larger than 0.4