I will provide some perspective for you on why she did what she did. She is certainly being selfish, but I don't think her intentions are completely horrible; from what I read, she didn't say anything too harsh about you. She just re-framed the situation to make you completely wrong. It sounds like your argument was not resolved in her eyes, and she needed to vent. Hell, we'd all feel better if we could frame a situation in a way that made us out to be the perfect hero and everyone would say, "wow! He sure was being difficult, he is so right". That's what she wanted to hear, so she framed it in such a way to prompt that response from her mother. It's a terribly selfish thing, in that she was willing to compromise your image to hear that validation. Essentially, she was being immature and selfish. These are qualities we all inherently possess, and it's easy to improve these qualities if we are willing to work on them.

It seems to me that your SO hasn't learned how her actions can affect the relationship and the way her family views you. This behaviour is very typical of much younger people... For whatever reason, your girlfriend never learned any better. You had an argument with your girlfriend, and it's clear that she felt guilty about how she has treated you, but also frustrated about the way you framed your request. She walked away from your argument feeling as though her emotions were not valid. You two need to sit down and figure out ways that you can both walk away from these discussions feeling better about things; I think you would both benefit greatly from learning to discuss these things in a more constructive way and acknowledge that the feelings of the other person are completely valid. If she knows that her feelings are OK and valid, she is much more likely to work on a solution with you, and much less likely to go seek validation from her mom (or anyone else) in her immature and selfish ways.

I would also request a firm "no badmouthing each other to anyone" rule. When an issue arises, it's not "me vs. you"; it should be "us vs. the problem".