I'm a long time lurker, and this thread motivated me to register. I flew in all categories of aerobatic competition between 1981 and 2006 in a Decathlon, S-2A and S-1S. This year I would have got back into it at the Advanced level in my new-to-me monoplane if it weren't for Covid. My next and last plane likely will be an RV-8.
I think what is missing from this thread are specific statements about what problems are solved by the emergency recovery techniques. It may be difficult for people to choose what works best for them or to justify spin training if they don't understand what can go wrong.
No one has described the trap in the standard recovery method, PARE (throttle off and ailerons neutral, rudder opposite the yaw, then forward stick for an inside spin). If you are too aggressive with the forward stick it is possible to transition smoothly with no hesitation into an outside spin that will not stop because the opposite rudder is now the correct rudder, the yaw continues in what can appear to be the same direction and the forward stick keeps the plane in the outside spin.
To apply the appropriate PARE recovery for an outside spin that started inside you first need to recognize that is what happened. If the original inside spin had the plane nose down beyond roughly 60 degrees it is possible to miss the change to inverted if the outside spin is also steep. You may think this should be hard to miss, as should the change to negative g, but I know of two crashes that were observed by very experienced aerobatic pilots and this was almost certainly the explanation. In one case, at Rocky Mountain House, AB, a prominent aerobatic coach talked to the pilot all the way down and couldn't convince him to apply the correct anti-spin controls in time.
There is an illusion that can make it hard to identify the direction of yaw in an outside spin. The spin axis points to a spot on the ground that is normally hidden by the fuselage in an upright spin, and may be hidden by the top wing in an outside spin in a biplane. You see the correct direction of yaw if you look along the cowl, but if you look too far from the nose (above the wing in a biplane) you see the ground turning in the opposite direction.
In a monoplane the chances of mistaking the direction of yaw are smaller than in a biplane. However, under the stress of an inadvertent spin that fails to stop, you might not reconsider which rudder and elevator are needed if you do not know that the plane can transition smoothly from an upright to an inverted spin. An additional issue when using PARE is that to be certain of recovery you must get the sequence of rudder and elevator reversals correct to recover from an outside spin that started inside.
The Beggs-Mueller technique (snap the throttle to idle, apply rudder opposite to the yaw and let go of the stick) makes it unnecessary to figure out whether a spin is outside or inside. Most descriptions of the Beggs-Meuller technique emphasize looking just above the nose to determine the direction of yaw.
The Finagin technique (throttle off, forcibly neutralize all controls and confirm visually) eliminates the need to determine the direction of yaw or the g direction.