Yes, that's correct. But, even though the engine and prop are still turning at the same RPM, with a constant speed prop the change in pitch allows the engine to operate more efficiently for a given RPM. So because of that a variable pitch prop is like a transmission in a car as Tom alluded to.
The Continental GTSIO-520 in the Cessna 421 is the most common one I can think of. The prop is geared lower than the engine so the engine can spin faster, which more engine RPM=more horsepower.