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You have a right to write them a letter explaining that you disagree with the facts in the reprimand and supply your own facts AND EVIDENCE to support what you are saying. You can ask the employer to reconsider based on the new evidence and facts presented. But I am afraid that would be the sole recourse you have, to just send in your letter and it is up to the employer as to whether they agree with your side or not and if they disagree they would not change the reprimand.