For a series that can spark a 'cross-the-board range of opinions on gender, feminism, and androgyny, one needs to look no further than Kaichou wa Maid-Sama! (2005).
Kaichou wa Maid-Sama! (literally, The Class President is a Maid!) is exactly what it says on the tin: By day, the heroine Misaki is the tough-as-nails president of her school's student council, to whom all the male students must bow down. But after school, she's a mild-mannered cosplaying waitress in a maid cafe, who must who cater to her male customers' every whim.
Obviously, a strict framework of gender identity is very potent in this series... while at the same time there are ventures by many different characters to cross those lines, the acceptable limits of which are constantly tested. One of Misaki's classmates, Aoi Hyodo, embodies this test perfectly.
Personally being a fan of the tsundere archetype, I began watching with high hopes for the series and its strong tsundere lead. Misaki has been praised by some as a feminist role model, but I honestly wasn't too impressed with the execution. Her self-proclaimed hatred of men certainly isn't a defining trait of a feminist, and despite her strength she feels overshadowed by her princely love interest Usui more often than not. I feel as if the writing lets too many gendered tropes of shojo romance get in the way of good characterization.
Regardless, even Misaki, who maintains a strict dividing line between the sexes in one way or another, can convincingly cross-dress and and embody androgyny when her job calls for it.
More explicitly, the character of Aoi Hyodo is the story's token gender-bender. Aoi's hobby is modeling in female clothes. He dons a blonde wig, borrows his aunt's dresses, and garners a strong following as an Internet idol under the name Aoi-chan. Like many gender-bending characters, his physical features are already naturally androgynous, but his transformation is still quite drastic.
Aoi's range of interests very much reflect a stereotypical female gender -- fashion, cute things, and riling up boys. Unlike others with similar hobbies, he is not portrayed as homosexual, yet he enjoys the fact that so many boys fall for his female persona. He goes to great lengths to tempt Usui, the one boy who doesn't, because otherwise he feels insecure in his cuteness.
Because throughout his childhood he has been bullied by boys for his girlish appearance and habits, turning the tables to make them swoon over him is what helps him feel more in control. For an androgynous character, he doesn't have a very gender-free mind. Aoi tends to have strong negative gendered opinions, not only taking out frustration on boys, but also criticizing girls like Misaki for lacking sufficient girlishness. Nevertheless, his "cute" interests are genuine and an important part of his identity.
Aoi faces rejection from his father and his second aunt Nagisa, who kick him out of the house, disapproving of his feminine ways and demanding that he "man up." At one point, after a great deal of fighting, Nagisa gives him an ultimatum that wages an end to his cross-dressing career. At this point, an interesting challenge is proposed: He'll be allowed to continue dressing as a female, only if he can prove his manly determination in a no-holds-barred game of volleyball.
The association of sports with masculinity is a classic one... and a rather stupid one, especially considering that Aoi's teammate is a girl and his original coach was a woman. Nagisa herself is the one who trained Aoi how to play volleyball, which may speak to double standards of acceptable gender expression. Regardless, the test put forward has clear intentions. He is allowed to express femininity, as long as he retains some masculinity to go along with it. Even where gender roles are enforced and full transgression is frowned upon, perhaps a certain type of androgyny (a mix of the two) is acceptable.
Not to say, at all, that this is the ideal-- but it's an interesting insight to how the standards of this situation work. It's worth thinking about how the real world compares: Is it more acceptable to be androgynous, or to fall at one end of the spectrum? Obviously, the answer varies across cultures and contexts.
On one hand, the preservation of a gender binary in an individual who is transgender (transitioning fully male-to-female, or female-to-male) sometimes makes things easier for supporters to grasp, as opposed to a more complicated, nonbinary identity that refuses to rest under either label. The "test" that needs to be "passed" is one of whether they can fully live up to a single role in society. On the other hand, it would be absurd to insinuate that transgender people who can pass as their preferred gender have it easy, while people who resist extreme labels may not be as actively questioned.
Furthermore, there are many contexts in which androgyny is viewed as attractive and fashionable, though the standards for androgynous beauty are rather narrow in the long run. There are certain types and balances of gendered traits that are considered acceptable, while others are viewed as ridiculous. Further furthermore, gendered personality and performative traits are often a separate issue from gendered physical traits and visual presentation.
Of course, the uniting problem with all of the above points is that they put value on an outside party's perception of gender over the individual's themselves. There is no reason for a test to exist at all, as there was no for relatives to be policing just how feminine Aoi is allowed to be, not even as a matter of compromise.
There are so many ways in which masculinity and femininity can combine that making assumptions about what's acceptable or what's the norm is nigh impossible, because experiences are so different for individuals.
People can be many things at once -- a leader and a servant (a president and a maid) or a boy and a girl... It's futile to get hung up on the levels, since everyone is already a mix of both.