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I'd remove the new starter and bench test it or take it back to where you got it and have them do it. Sounds like the motor is spinning but the bendix isn't moving and the the piñon gear isn't engaging with the flywheel. Your new starter is just a rebuilt unit and could be defective. This just came up in another thread. When your starter goes out, you're usually just as well off taking it straight to have it rebuilt yourself. The only downside is it might take a few days to get it back.
I don't think you can install it backwards. The backside of the housing would contact the flywheel and not allow it to locate properly.
You should be able to turn the gear by hand and extend it out like its engaging. Have you checked the trigger wire for power when you turn the key to start? If the starter works on the bench, that to me would be the next logical thing to check.