I took one (only one) and I think you can do it that way. Chords are easy to make and pickin' notes up and down the neck shouldn't make a difference. (It would be impossible to do this with five string banjo btw, b/c part of the neck is cutaway and the nut is on the neck).Mandolin seems like you could play this way, and there are certainly enough guitar players,known and unknown, who have done it.

Purists would argue you should restring, but you will be joining the small but elite crowd of lefties who play 'upside down and backwards,' including Bill Staines, Libba Cotton, Jimi Hendrix and Babyface.

Go for it!