Are you looking at a planar microcavity or is it a structured one, e.g. a micropillar?
Generally speaking, the TE and TM modes are the linear polarizations with either the electric or the magnetic field oriented perpendicular to the plane given by the normal vector of the microcavity plane and the wave vector of the light field. Therefore, the TE-TM splitting will obviously be slightly different for each value of k along the dispersion and it is 0 for normal incidence.
So you need polarized light and a half wave plate to rotate the polarization of the light field. One possible way to do this is to perform angle-resolved reflection or transmission measurements using polarized white light. If you do not know, at which angle you need to put the half wave plates, just rotate them until you find the lowest and highest energies for some given angle. These should appear at orthogonal angles. The splitting is, however, usually not large. Depending on the material, it may be a feww 100 microeV, though.