Interesting question, as in aikido we use the sword to study techniques and movement rather than for dueling. If we do left and right techniques, why not do left and right aikiken, irrespective of what the koryu folks do? Seems I have heard somewhere that Chiba Sensei does do this. I sometimes do it, and I'm even worse using the sword left handed than right handed.
WRT writing, one of my first memories of school was a teacher trying to make me write right handed. Being naturally perverse, I now write almost exclusively with my left hand. But I use a hammer right handed, so I'm probably ambidextrous.
When I played around with shodo, I tried the brush both ways, and there doesn't seem to be as much difference as in western writing, where the forearm rests in the paper and smudges what you have just written with your hand curled around to get the letters slanting the right way. Long live the keyboard!
Techniques? Seems like one side is as easy as the other. So maybe sword practice is not that important after all.