‘‘It is startling that the Commissioner has damaged the careers and reputations of the Players on such scant, contradictory and incredible sources,’’ the NFLPA said.

A three-member appeals created by the NFL’s labor agreement vacated the initial suspensions on Sept. 7 and told Goodell he needed to clarify the basis for his rulings. The panel noted that punishments should not have anything to do with cash paid out of the Saints’ pay-for-performance pool, because an arbitrator other than the commissioner is supposed to handle such salary cap violations.

The latest legal filings point out that Goodell repeatedly mentioned pay-for-performance allegations as part of the basis for the initial punishment, and that the commissioner’s decision to maintain similar suspensions highlights the lack of fairness in the process.

Saints coaches and players have acknowledged the existence of a pool that both fined players for penalties and offered rewards for big plays, including big, non-penalized hits that may have resulted in opposing players leaving games for a play or longer.

Goodell has stated that in their acknowledgement of the pool, the Saints have admitted they encouraged hits that were shown to have injured opponents. Regardless of intent, Goodell said, such a program is intolerable because it sends the message that hits that hurt opponents deserve a reward, and that can affect how players on team approach subsequent games.