cutadapt should generally be able to handle the mate pair case though you need to be careful configuring it correctly for the purpose.
You might want to consult the cutadapt command line tool documentation on top of the help provided in the Galaxy tool interface and read through this old SEQanswers thread including that Illumina pdf linked to in the original question there.
In any case, you will want to inspect the output carefully, possibly also after mapping (if read mapping is one of the steps in your envisioned analysis).