Just a slight correction to your statement, rodeo's are a cross between backwards and Lincoln set. Think about it this way, for a backflip you thrust you hips straight forward to set your feet right over your head, a lincoln is set with your hips going straight to the side. A rodeo would be in between those to, your hips go forward and to the side and this ends up being a rodeo.
Flatspin is set like a rodeo except you block your hips a bit making it flat. They are not set like a backflip or you'll end up doing a crooked backflip. A true flat three is closer to an under rotated flat 5 than a rodeo 5. Here's a video of a true flat 3 along with the one that was linked earlier in this thread:
https://www.newschoolers.com/membervideo/316661.0/True_flat_3__to_1_ski_landing?s=126042&t=2&o=8
Lincoln 180 you throw your hips completely sideways then use your shoulders to spin the 180.
Hope this helps!