In a single-speed capacitor-start motor, the start winding, capacitor and switch are connected in series, and that string is in parallel with the run winding. That much is standard. But for a two-speed motor, a single start winding could only have the right number of poles for one of the speeds.
You could have two separate start windings for the low and high speeds with the correct number of poles on each, but that is rather wasteful of cost and size. It would also need separate centrifugal switches and capacitors for both start windings to avoid back-feeding the unselected run winding via the two start windings.
A more practical arrangement would be to use the centrifugal switch to force the motor always to start in high-speed mode. The motor has start and run windings for high speed, plus a run winding for low speed. The centrifugal switch has two sets of contacts: an N/C contact that controls the start winding in the usual way, plus a changeover contact that diverts the low-speed input to the high-speed winding until the motor reaches low speed.
If the high-speed input is energised, the motor starts and runs using the high-speed start and run windings. If the low-speed input is energised, the motor starts using the high-speed start and run windings, then as it reaches low speed the high-speed start and run are disconnected and the low-speed run engages instead.
A search of patents might reveal whether makers have implemented this system on pool pumps. TBH I've never seen a motor wired like this and just worked it out myself, but it's so logical and straightforward that I would be very surprised if something extremely similar was not used for this application. If I get time tomorrow I will search the patents.