WB: "And in some types this is how to execute a crossover spin depending on how fast and how far past neutral you go."
Interesting, in my experience that is not the case -unless you still have yaw, which should not be the case if you followed the R instruction in PARE. If you don't concur, can you explain why the spin flipped?
Cheers
Sadly students do not do the R bit. They leave the opposite rudder on to long, or they do not centralise correctly. In your example leaving full rudder opposite on (this will shortly after the balance point induce yaw in the opposite direction) and pushing the stick past neutral (albeit aggressively) will send one type of aerobatic aircraft I fly into a crossover spin. Some types will tolerate slight rudder mishandling others will not.
As a basic concepts go, PARE has validity with stable run of the mill tourers. However there are so many variables with aerobatic aircraft any cover all spin recovery technique may induce its own hazards. There is nothing to beat flying with an experienced instructor on type.