Smick – The Gemorah in Yevamos Daf 117a talks about this type of hatred:
“And R. Judah? – Because one can well understand why a mother-in-law should hate her daughter-in-law, since the former believes that the latter squanders her Savings, but why should a daughter-in-law hate her mother-in-law! Similarly one may well understand why a husband’s daughter hates her father’s wife, since the former believes that she is squandering her mother’s savings, but why should a father’s wife hate her husband’s daughter!
[this is the true explanation]: Why does a daughter-in-law hate her mother-in-law? Because the latter reports to her son all that she does. [Similarly]
So you see the Gemmorah brings a good reason why a MIL hates her DIL, because she will eventually, if not now, squander all her and her husbands’ hard earned money! Now acc. to R. Yehudah, the DIL hates the MIL because she snitches on her. Well the solution to this is don’t do things behind your husband’s back and there won’t be anything that she can snitch on. But the main thing is acc. to the Rabbonim, just because she hates you, even though there is a tendency to hate back, you don’t have to. If you break this cycle and not hate her back, eventually she will start liking you (Kmayim Ponim El Ponim)! And this is the way of our Torah!