Valaquenta: "But when they desire to clothe themselves the Valar take upon them forms some as of male and some as of female; for that difference of temper they had even from the beginning, and it is but bodied forth in the choice of each, not made by the choice, even as with us male and female may be shown by the raiment but is not made thereby."
What this means is, gender is decided by fëa (spirit), not body.
If Varda wanted to don the body of a man, she could. She'd still be female. It isn't the genitals that decide, it's the 'difference of temper' that's been there from the beginning. And that's canon. Conceivably a fëa could get clothed in a body of a sex that's different from the gender of the fëa. Ainur don't have those issues because they aren't created with physical bodies, they choose their bodies and by extension, their sex as they would choose clothing, so of course they choose what feels good and fits right. If they don't like any clothing, they don't have to wear anything. (See my post about Ulmo.) Anyone who isn't one of the Ainur could experience this, though.
Trans, non-binary, and other gender non-conforming characters in Ëa exist and are valid!