We report the structural, magnetic, and thermodynamic properties of the double perovskite compound La₂CuIrO₆ from x-ray, neutron diffraction, neutron depolarization, dc magnetization, ac susceptibility, specific heat, muon-spin-relaxation (μSR), electron-spin-resonance (ESR) and nuclear magnetic resonance (NMR) measurements. Below ∼113 K, short-range spin-spin correlations occur within the Cu²⁺ sublattice. With decreasing temperature, the Ir⁴⁺ sublattice is progressively involved in the correlation process. Below T = 74 K, the magnetic sublattices of Cu (spin s = 1/2 ) and Ir (pseudospin j = 1/2) in La₂CuIrO₆ are strongly coupled and exhibit an antiferromagnetic phase transition into a noncollinear magnetic structure accompanied by a small uncompensated transverse moment. A weak anomaly in ac susceptibility as well as in the NMR and μSR spin lattice relaxation rates at 54 K is interpreted as a cooperative ordering of the transverse moments which is influenced by the strong spin-orbit coupled 5d ion Ir⁴⁺. We argue that the rich magnetic behavior observed in La₂CuIrO₆ is related to complex magnetic interactions between the strongly correlated spin-only 3d ions with the strongly spin-orbit coupled 5d transition ions where a combination of the spin-orbit coupling and the low symmetry of the crystal lattice plays a special role for the spin structure in the magnetically ordered state.