Yes a battery can be reverse charged. However, the battery must be almost completely discharged for a charger to do it otherwise the charger will overcurrent as the battery voltage will be added to the chargers potential causing excess current to flow. In fact, the way to correct the reversed charge is to completely discharge the battery and recharge it correctly. BTW, the reverse charging and the rereversal lowers the capacity of the battery so you might have the cranking amps of the battery checked.
As to the damage done, connecting the battery to the car’s electrical system with the polarity reversed would cause an immediate short circuit through the alternator diodes. If the electrical system of your car is sufficiently protected there is an alternator fuse or system fusible link that will open to protect the electrical system. During the time it took the fuseable link to melt out, there would have been ~1.5 volts negative on the positive side of the system. If you have an alternator fuse that blew, then you might have had a negative 12 volts on the positive side of the system. If so you may have colateral damage to any module that is wired ‘always hot’ i.e. ECM memory, radio memory, air bag SRS controller, etc. I assume you did not connect the battery with the ignition switch ‘on’.
I hope this helps.