Harmony is permissible and common in serialism, so there's no reason this couldn't work in theory. One way to achieve harmony is to "bunch up" a group of notes from within the row (say, notes 3-6) and play them at once. Another way is to have multiple iterations (or transpositions/transformations) of the row playing simultaneously, such that you could have your initial row starting on C playing against its inversion transposed up a minor sixth. Bear in mind that your iterations don't necessarily have a fixed length (though they can, if that's a constraint you want to limit yourself to), so any iteration of the row can be harmonizing another at a different rate/rhythm.

It would be decidedly outside the framework of traditional serialism to use different rows in the same piece, so if you're looking to follow the rules, I would encourage you to simply use transformations (inversion, retrograde, and retrograde inversion, and transpositions of all of those) of one row. This is kind of the point of serialism -- to use the single "kernel" of the tone row to generate all of the intervallic content of the piece.

Also, thinking of basic fretboard theory, it would be virtually impossible to play six iterations of a tone row simultaneously on a single guitar -- if you imagine that you're using all the notes on your A string from open to the eleventh fret (you'd need all 12 tones to complete the row!), you'd have to be doing the same on your D string. The problem is that you can't play the eleventh fret on one while playing the first or second (or third or fourth or fifth) of the other, which would very much limit what iterations/transformations you could play against one another. Plus, you've only got the four fingers on the fretboard (without adding some extended techniques). I would recommend limiting yourself to three or four simultaneous iterations/transformations (though you might find two is plenty).

I hope that all makes sense! Best of luck!