erikv,
You are correct in believing that the aircraft in a spin pitches up continuously, although there may be variations in the amplitude of the pitch-up.
To visualise this, imagine a light aircraft in a spin, and isolate pitch by "suspending" the aircraft from the tip of its out-spin wing.
The aircraft is now in the "spin" with its wings being vertical, with no yaw (other than it is descending) or roll. In order to follow the flight path of the "spin", (which is now like riding the "wall of death" in a fairground) the nose has to pitch up constantly.
Also consider the recovery action, whose main component is "stick forward until the spin stops". It is felt that pilots who have had difficulty coming out of a spin have been too ready to try something different to the standard recovery action, when they should have held the stick forward "until the spin stops".
I follow the rule of thumb - if it is difficult to get the aircraft to spin, it is likely to be difficult to get it to recover.