An angular analysis of the B0 → K∗0 (→ K+π−)µ+µ− decay is presented. The dataset corresponds to an integrated luminosity of 3.0 fb−1 of pp collision data collected at the LHCb experiment. The complete angular information from the decay is used to determine CP-averaged observables and CP asymmetries, taking account of possible contamination from decays with the K+π− system in an S-wave configuration. The angular observables and their correlations are reported in bins of q2, the invariant mass squared of the dimuon system. The observables are determined both from an unbinned maximum likelihood fit and by using the principal moments of the angular distribution. In addition, by fitting for q2-dependent decay amplitudes in the region 1.1 < q2 < 6.0 GeV2/c4, the zero-crossing points of several angular observables are computed. A global fit is performed to the complete set of CP-averaged observables obtained from the maximum likelihood fit. This fit indicates differences with predictions based on the Standard Model at the level of 3.4 standard deviations. These differences could be explained by contributions from physics beyond the Standard Model, or by an unexpectedly large hadronic effect that is not accounted for in the Standard Model predictions.