I’m literally just moving flat rectangles around inside another flat rectangle, with each rectangle as a group, and I’m still getting nearly-but-non-perpendicular angles. How is this possible? Length snapping is off.
You would need to show us the model for us to be able to help.
It is possible you are looking at your model with the camera set to parallel projection and you aren’t looking straight at the faces.
I agree with @Box , post a file or we can only guess. You can verify by turning edge color to By Axis in the Style panel, to help verify.