Haying hillsides

My fields are flat, which is rare in Pennsylvania. Three neighbors have offered me their fields, and two are hilly.

I have never rolled a tractor, and never plan to. I have mowed some steep side hills with a brush hog, and I have turned downhill when I didn't like the way ot feels. No problems, i can mow up and down, or just skip it.
Haying is different. I want long straight stretches, and I have different implements.
How do haying implements affect stability on side hills?
I have an offset mower, its a rotary scythe. Its low to the ground,and I think it could only push me by the drawbar, and maybe make me slide downhill.

I also have a round baler, a Vermeer 605c. That thing looks tall enough to roll. Do round balers ever slide sideways downhill, or do they have enough traction to track straight?

Has anybody ever rolled while haying?

Any tips to keep me upright?

These hills are not as steep as some that I see folks haying, but they are steep to me. The good thing is they level out as you go down hill, so I can turn downhill and not hit anything.

Haying hillsides