The nuclear magnetic dipole moment of ¹⁸⁶Ir has been deduced from a study of the gamma rays emitted by ¹⁸⁶Ir oriented at low temperature in an iron host. The deduced value of 3.88(5) mu _N compares favourably with recent NO/NMR results. The measured magnetic dipole and electric quadrupole moments are shown to be consistent with values calculated assuming the spin-5 ¹⁸⁶Ir to be a mixture of K=0 and K=1, with the interference term between the two components contributing about half the value of the magnetic moment. Multipole mixing ratios of ¹⁸⁶Os gamma rays are deduced and compared with results of band-mixing calculations.