I had my saddle slip to the side while I was riding up a steep hill. It was very scary. I jumped off and fixed it. The saddle was a little too big for my horse so it slipped. I purchased another saddle that fit her better and I didn't have that problem, until I tried putting a thick pad under the saddle. When I mounted the saddle slipped to the side, so I removed the pad and everything was fine.
When I was young, I had a pony that always "puffed up" so I developed the habit of cinching up and then walking the pony for a few minutes. The cinch was always loose and I had to tighten it before I mounted.
I hope it doesn't happen to you again.