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Does your starter have a clutch type bendix on the gear? If so, it is possible that the rollers are gummed up and will not "hold". A temporary fix is to flush out the bendix with starting ether to see if that remedies the situation. If it does, the motor can be used for a while yet, but the starter bendix needs to be replaced. Otherwise following through on intial line of thought- Are the teeth intact on the flywheel too? Are they worn to any depth? Sometimes the teeth may be worn just enough to miss engaging with the starter. As for things being counter intuitive, remember that the motor will turn opposite the direction of the engine, and that the engine is "trying not to turn" (An adaptation of one of Newton's law's of motion without the math), so it will seem that the starter is needing to turn opposite the direction that it actually does. The way every starter motor is put together (at least the ones I have encountered), will assemble only one way, and will only rotate one direction regardless of the battery polarity applied. If the starter was working with the engine, and now it is not, suspect the flywheel, or the starter bendix. Depending on how the starter ring gear is attached to the flywheel, it may actually have broken loose, and may get pushed out of the way by the starter gear, or got pushed out of the way.
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