Is the same action being prohibited for each sex, or is the prohibition for women different from the prohibition for men?
Based on the analysis of the verse’s terminology offered above, it seems most likely that the text is prohibiting two different things. The second part of the prohibition is clear: men cannot wear a woman’s (outer) garment.
Rabbinic sources also expand the prohibition. Rabbi Eliezer ben Yaakov (Sifre 226 and the b.Nazir 59a) understood it to extend to jewelry and other female adornments, while others extended it to grooming behaviors associated with women, such as wearing particular hairstyles, plucking out white hairs, or shaving or cutting hair from certain body parts (b. Shabbat 94b; Makkot 20b; Nazir 59a).
The first part of the prohibition may be stating that women are not to have on them any item that is associated with men, which would include most or all weapons, and certain tools, as well any clothing items considered “manly.” Thus, while the two parts of this prohibition are related, they are not identical.