Air carrier flight crew reported a loss of aircraft control during gate tow in procedure.

Narrative

We were being towed into Gate X and were having communication issues with the tug driver due to equipment issues (their headset or plug). We were informed he could hear us though we could not hear him. When we concluded that we were in position and that the tow in process had been completed the Captain asked the tug driver if they wanted brakes. The tug driver responded not to set the brakes and that he needed to move us. Following some extended movement backwards we heard a swift set brakes". We then felt another capture of the nose wheel and a command to release the brakes with a movement forward this time. Upon completion of the tow in and checks we assume we may have rolled back; but no confirmation of this at this time. Following deplaning we spoke to a Maintenance Technician and he confirmed that we had rolled backwards more than six feet; but we were unable to speak with the tug driver because they had already left.Cause: Being released from the tug without the aircraft being secured in addition faulty tug radio equipment causing communication issue Suggestions: Not being released from the tug till the aircraft is secured. Maintaining standard operating procedures with functional communication equipment"