Sexism does go both ways. In fact there is a lot of animosity between the genders where one blames the other for their (personal) unhappiness in life.
Some women are "man haters" and some men are "misogynistic".
A problem arises when one automatically lumps all men or women into a category instead of recognizing each person is an "individual".
Taking a "gender shortcut" by stereotyping people simply based upon their gender is being (lazy).