Measurements are presented of the polarisation of W+W- pairs, and of CP-violating WWZ and WWGamma trilinear gauge couplings (TGC). The data were recorded by OPAL during 1998 with a total integrated luminosity of 183 pb^-1 at sqrt(s)=189 GeV. The measurements are performed through a spin density matrix analysis of the W decay products. The fraction of W bosons produced with longitudinal polarisation was found to be sigma_L/sigma_total = (21.0 +- 3.3 +- 1.6)%. The joint W boson pair production fractions were found to be sigma_TT/sigma_total = (78.1 +- 9.0+-3.2)%,sigma_LL/sigma_total = (20.1 +- 7.2 +- 1.8)% and sigma_TL/sigma_total = (1.8 +- 14.7 +- 3.8)%. In the CP-violating TGC sector we find kappa_z = -0.20 +0.10 -0.07, g^z_4 = -0.02 +0.32 -0.33 and lambda_z = -0.18 +0.24 -0.16. In each case all other couplings are set to their SM values except those related to the measured coupling via SU(2)_LxU(1)_Y symmetry. These results are consistent with SM expectations.