I always thought the title ‘The Sword Who Cuts the Heavens’ represented the MC’s capabilities as a swordsman and implied that they were graced with the skill to cut even those of divinity. Though, with the knowledge that ‘Jigoku Ittō-ryū’ translates to Hell’s Release Style, suggests a slightly darker significance to the title.
The concept of divinity of Emperor’s is something to take into account as well. Shintō, Japan’s traditional religion, proposes that the Emperor descended from the gods and was also in contact with them.
Now, I’m not sure if Hyuga follows the belief that their ruler is touched by the gods, but the MC’s title might just be foreboding something…
Also, I don’t think Satsuma is a demon per say, but I wouldn’t put it past the Emperor to pull some strings in order to stay in power. Though if the MC manages to survive all demons and Satsuma is the final mastermind, my ronin will gladly slay the damn lion. I never appreciated being lead around on a leash for the benefit of his hierarchy and I still don’t.