I see. The only problem I see here is that if you straight invert the values for the mouse input when upside down, it will cause a weird feeling and particularly when on the edge. I wonder if you should not rotate Math.PI on the horizontal axis when you pass the inverted point, and do it with an offset/deadzone and a slow animation (as if the character/astronaut would rotate to put himself in the upright position relative to the object). I guess this would feel more natural and would make more sense then simply inverting the inputs? My opinion only. I suppose only a test will tell.
Edit: I just did read your post again and I think what you are refering to is ‘leaning’. I’m just wondering how you can turn and lean with a natural feeling simply with the mouse. May be you should assign keys to lean?