When I brake at speed rear end moves sideways

Take a technician for a road test after basic safety checks before operating

The defective part may be your rear sway links - but until you have a qualified mechanic check it out, it isn't safe to drive.

There is some question about the symptom, can you be feeling a brake related pull ? like one side is ineffective, making the other seem more effective ?

Could it be the rubbers that are joined to the bar above the diff

Sure, if it controls lean and sway and they fell out. or if it locates the diff attitude and position. Why, what are yours like ?

The middle rubber on top of diff looks worn and it squires when you push car side to side

Does not sound unsafe so you can take it to get looked at. Does not sound like it would cause a concern either but I have not driven it. When you experience this feel like the rear axle moves, can you try to give that some more description ? We need to know if you have an issue with lugnuts or brake or axle not assembled or some tire pressure issue or what have you ? This is why i wanted to inspect before road testing. Can you jack it up ?

What happens when holding the brake at the curb, and shifting into reverse and then into drive ?

have the lower control arms looked at the bushings could be worn and the sway bare linkage also. this happens when you let off the gas going down hill or the cruze lets off?

There is no problem when the car is going straight or turn. It's only when I'm stopping it, it goes sideways. Please, can anybody explain why?

Did you ever figure this out? Mine does it and I can’t figure it out.