What is the verb for when a person criticizes you back instead of accepting the criticism?
Or more modern name than one proposed by nene: "whataboutism".
This is called the
Tu quoque fallacy,
or the appeal to hypocrisy. It's an attempt to discredit a statement by saying that the speaker of the statement is guilty of the same problem.
As with all fallacies of irrelevance, sometimes the situation is relevant. This is more familiarly referred to as 'the pot calling the kettle black'.