Experimental investigations of two schemes for polarization-direction-insensitive spectral conversion by use of four-wave mixing in a semiconductor optical amplifier (SOA) are reported. One scheme that utilizes an SOA with a linearly polarized signal and a single copropagating circularly polarized pump is, as far as we are aware, the simplest such scheme that has been reported to date. A theoretical analysis of this novel scheme is also presented. Experimental investigations of that scheme are compared with measurements of the performance of a previously proposed scheme, which involves the use of a single SOA and two orthogonal counterpropagating pump beams. We observe that polarization-direction-insensitive conversion is far more efficient in the new scheme described here. Within the observed detuning range, the difference in efficiency between the two schemes is found to increase from 6 to 18 dB as the detuning is increased. The corresponding increase in the difference of the signal-to-background ratio between the two schemes is from approximately 2 to 16 dB.