The azimuthally rippled cavity for a large orbit, co-harmonic gyro-multiplier, designed to operate at the 2nd and 4th harmonics, at frequencies of 37.5 GHz and 75 GHz, respectively, has been numerically and experimentally confirmed to be insensitive to the polarization of quadrupole, TE2,n-like modes, including the 2nd harmonic operating mode of the multiplier, a cylindrical TE2,2-like waveguide mode. To test the cavity with this mode required the design, construction and measurement of ripple wall mode converters, converting the cylindrical TE2,1 mode to the TE2,2 mode. These were designed to operate at a central frequency of ~37.9 GHz, with predicted mode purity of better than 85%, and 3 dB bandwidth of 161 MHz. The constructed converter had a central operating frequency of 37.7 GHz, with S-parameter measurements used to infer suitable mode purity, and an operational 3 dB bandwidth of 50 MHz. This has allowed farfield phase measurements of the corrugated cavity to be conducted, where the orientation of the geometry to the polarization of both the TE2,1 and TE2,2 modes was shown to have no effect on the dispersion.