A lot of things have changed over time but if we were talking about historical accuracy, tachis were tied and hanging from a samurai's waistband with the blade facing down while katanas were simply inserted or tucked inside the waistband with the blade facing up. Granted there were vast differences in how samurais wore them—and in recent examples—how Isaac and I wore them, but let's just say Isaac was wearing his katana in the traditional sense while I was doing a mix of both.
It was because I had my katana tied tightly to my war belt, not hanging down, and the blade was facing downward.
There wasn't any particular reason for the way I was doing it because I even used to wear my wakizashi—that was now on Kaley's hips—horizontally on my back, and it was partly for coolness but more along the lines of comfort and just how it felt right.