The wording of this clause is merely a figure of speech, an idiomatic usage, and should not be taken to refer to the person himself or herself. The Torah's concern in this verse, then, is not with creating or reinforcing gender differences per se, but in preventing gender associations of clothing or possibly body hair from being used to deceive others for purposes leading to sexual immorality. The key here seems to be deception for illicit purposes. Indeed this law appears in Deuteronomy in the context of laws against deceit. At the movie theatre the teenage women selling popcorn are wearing suits and ties.
Jeans are common to all genders as are sweaters and jackets.
It is much more common for genuine intentional transvestites to be accepted publicly. A person's gender can less easily be judged by their clothing, and many people like it that way. This discussion is meant to be informational only, not determinative of Jewish law for this verse or the issue of cross dressing generally. Persons of the Jewish faith should abide by the legal interpretation of the verse as it is understood by their local community as explained by their rabbi. If you would like to learn more of what the Torah teaches on this and other subjects, go and study.