I did some spin training in a T67 with 160hp and a fixed pitch prop where I believe we were within the CG limits. On one upright spin to the left I put in full right rudder and was then in the pause before moving the stick forward but before I did that the spin 'instantly' changed to one to the right. I was thinking to put in left rudder but my instructor very quickly took control and recovered, I guess he wanted no delay. So the lesson learned was to neutralise the rudder as soon as the rotation stopped and perhaps be a little quicker moving the stick forward.
I now fly a Pitts and there have been cases of people being in an upright spin in one direction crossing over to an inverted spin in the other direction whilst trying to recover and being unable to recognise the change before crashing. So again the lesson learned is to neutralise the rudder as soon as the rotation stops and perhaps be NOT so quick in moving the stick forward.
It seems all types have their peculiarities in the spin, at least in the Pitts the Mueller/Beggs method works when you are unsure what to do.