Planes of symmetry are often identified by the existence of pure cross-plane polarization. However, this type of polarization can occur without the plane being a plane of symmetry. Planes that support cross-plane polarization are called "decoupling planes", since the system of three coupled linear equations in the direction cosines of the polarization Vector decouples into a single cross-plane equation and a coupled pair of in-plane equations. Only if the direction perpendicular to a decoupling plane is a "longitudinal direction" (i.e., if in the direction there are pure P- and S-waves), it is a plane of symmetry. Without the observation of the associated longitudinal direction, a "raw" decoupling plane might be mis-interpreted as a plane of symmetry. The plane perpendicular to the i-direction is a decoupling plane if in four-subscript notation all stiffnesses with a single subscript i vanish; the i-direction is a longitudinal di-rection if all stiffnesses with three subscripts i vanish. In media of orthorhombic or higher symmetry all stiffnesses with any single or triple subscript vanish; therefore raw decoupling planes can occur only in media of monoclinic or triclinic symmetry. In triclinic symmetry, two mutually perpendicular raw decoupling planes can occur. Decoupling planes intersecting under an oblique angle are possible if the stiffnesses satisfy a number of constraints.