This has been bugging me for a while now, but just so you know, katana are worn with the blade facing up, not down. There are other Japanese swords that look fairly similar (in particular tachi) that are worn blade-down, but katana are specifically forged in such a way that they are weighted to sit blade-up for the purpose of streamlining the initial drawing motion. That being said, early on the MC was generally shown wearing his with the blade up, but as the story went on it started to become inconsistent.
P.S. The scene after sleeping where the MC and Ryoko leave to go to Sakura Town for the "Small Strengths" quest triggered before the "My Purpose" quest was finished, when the former quest clearly states that you need to complete the latter before it will be safe enough to do that (as well as the dialogue from when Small Strengths was first acquired).