It is in fact possible to recognize that constructions of masculinity are damaging and that there is gender inequality that goes both ways without supporting the cesspool that is the Men’s Rights Movement.
The biggest problem with the term “misandry” is the false equivalency that these oppressions are remotely on the scale of misogyny. They are not. In fact, most of them are rooted in oppression of minorities, such as women or trans* people or queer people or people of color or poor people.
Of course sexism affects men — but it primarily affects men because it oppresses women. This is what I mean when I say (as I do, and often) that there is no institutional sexism against cis men. There isn’t. There is institutional sexism against women that creates awful, damaging constructions of masculinity based on societal distaste for femininity. Lest we forget, constructions of masculinity are damaging because femininity and associated characteristics are seen as bad. Standard constructions of masculinity and the pressure to conform thereof stems from misogyny, not some new brand of anti-male sentiment. The fact that sexual assault and domestic violence against men goes underreported is a direct result of this societal terror of “feminine weakness” in our cis men. Not to oversimplify an incredibly complicated piece of intersectionality, but this is in part why trans* people (especially trans* women) and gay men are seen as some sort of aberration by the more odious parts of our society — femininity itself, and all incarnations or perceived parallels thereof, is fucking terrifying. Why do you think Michigan just got so horrified by one woman talking about her vagina?