Measurements are presented of the polarisation of W+ W− boson pairs produced in e+ e− collisions, and of CP-violating WWZ and WWγ trilinear gauge couplings. The data were recorded by the OPAL experiment at LEP during 1998, where a total integrated luminosity of 183 pb−1 was obtained at a centre-of-mass energy of 189 GeV. The measurements are performed through a spin density matrix analysis of the W boson decay products. The fraction of W bosons produced with longitudinal polarisation was found to be σL/σtotal=(21.0±3.3±1.6) where the first error is statistical and the second systematic. The joint W boson pair production fractions were found to be σTT/σtotal=(78.1±9.0±3.2), σLL/σtotal=(20.1±7.2±1.8) and σTL/σtotal=(1.8±14.7±3.8). In the CP-violating trilinear gauge coupling sector we find κ~z=−0.20+0.10−0.07, gz4=−0.02+0.32−0.33 and λ~z=−0.18+0.24−0.16, where errors include both statistical and systematic uncertainties. In each case the coupling is determined with all other couplings set to their Standard Model values except those related to the measured coupling via SU(2)L×U(1)Y symmetry. These results are consistent with Standard Model expectations.