It is known that electromagnetic waves propagating in a magnetized cold plasma are not necessarily purely transverse waves. The existence of a longitudinal component implies that the normal to the polarization plane is not, in general, parallel to the propagation vector. The offset angle between these two directions is computed for each characteristic wave. Our investigation shows that the offset angle can be large under certain conditions. For a wave field that is a superposition of two characteristic waves, its polarization plane wobbles in space as the field propagates while its normal precesses. Several examples are considered, computed, and given to illustrate the 3D nature of wave polarizations.