Air carrier Captain reported ground crew failure on push back resulted in tow bar disconnect and uncommanded aircraft movement.

Narrative

We pushed back from Gate in ZZZ and as we were nearing the end of the normal push; the Ground [Crew] very urgently told me multiple times to set the brakes. I applied moderate pressure to the brakes and the aircraft came to a very abrupt stop. That's when I noticed the tug and tow-bar about 20-30 ft. in front of the aircraft. With the parking brake set; I immediately called the Flight Attendants to see there were any injuries or damage due to the abrupt stop. They indicated that everything was okay and there were no injuries or other problems. I then asked the Ground Crew to check over the aircraft before departing. They indicated everything looked good to them. We then continued with engine start and before taxi check. Prior to taxi; I checked with the Flight Attendants (F/As) one more time to make sure everyone and everything was okay. When they said it was; we taxied and took off and the rest of the flight was normal. When we were just starting descent into ZZZ1; the F/As called me and asked for more details because they were considering reporting injuries from the abrupt stop. I explained in detail what had occurred.Apparently; the tow bar was attached to the aircraft; but not connected (the pin was not in); so the tug was able to push the aircraft; but when the Tug Driver applied the brakes; the tow bar became disconnected; and the aircraft continued to roll backwards. This was an exceptionally dangerous situation; and the outcome could easily have been much worse. Another likely factor was that when we landed in ZZZ; the right thrust reverser failed to deploy; causing a maintenance light on the overhead. I discussed this with Maintenance Control and we had contract Maintenance come out and do the necessary checks. Part of this included pressurizing the A hydraulic system to cycle the reverser. I assume this step would require disconnecting the tow bar to avoid potential injuries from the nose wheel steering. I do not know how this was done or by who; but apparently it was not reconnected properly.