In the case of the Competition Pro (which my arcade stick is based on), the original 5200 stick acts as the centers. The pots are used in conjunction with the arcade joystick. Center the 5200 stick and the arcade stick is centered. If the 5200 stick is off center, the arcade stick will act as if it is too.
To my understanding, the pots range from 0-500... 250 being center. The Competition Pro is wired to allow a constant one-way connection for the analog signal. Move the arcade joystick in one direction, and the stick hits 500, move it to the other direction and it goes to 0. The stick at rest is at 250... center (provided the 5200 stick is centered).