In medieval Japan (and for the longest time into the modern age), Japanese children were taught to write, eat, and wield their swords with their right hands. In particular, etiquette demanded that samurai wear their swords on their left hip, as people walked on the left side of the road, and to wear on the right hip would have made it a bother for oncoming traffic.
Likewise, the kata in sword schools were designed with the idea that both participants were right-handed. To do them left-handed would obscure the lesson to be taught in the kata.
That said, some schools, as a tactical or training consideration, included left-handed techniques among their primarily right-handed curricula. To my knowledge, the only extant schools that still do are Yagyu Shinkage-ryu and Yagyu Shingan-ryu. Though there may be others.
Josh Reyer
Swa sceal man don, žonne he ęt guše gengan ženceš longsumne lof, na ymb his lif cearaš. - The Beowulf Poet