I will put this here for future reference for anyone searching on this topic. To reverse a single phase motor, you reverse the polarity of the start winding. In a properly marked motor, the run winding is A1 and A2 and the start winding is Z1 and Z2. You simply swap over Z1 and Z2 connections. The problem is identifying the leads if they or the terminals are not marked up properly. If the motor has a centrifugal switch, it switches the start winding out of circuit, so if you have one lead marked Z and another going to the centrifugal switch, they are the start winding. The start winding will have a higher resistance than the run winding when checked with a multimeter. Sometimes other letters are used (thanks to EU harmonisation) if in doubt check the windings resistance. If your motor is as full of swarf as the op's, best strip it and clean it out, as the centrifugal switch has bare live contacts on it, which will set oil filled swarf on fire in seconds, doing the lathe and the workshop a lot of no good!!
Phil
Edited By Phil Whitley on 15/01/2022 18:00:33