Let’s discuss Deuteronomy 22:5. Wording of this verse is unusual, because it uses the word gever - master or great person, instead of adam or aish - man, and also the word chli- an article or furnishing instead of say standard garment (bigdey) so I'm not certain what it may imply. However the translation seems clear to me (at least at the moment - i've never researched it) that is has something to do with the transvestite behavior. I'e a man should not wear shoes with heels, and a woman should not wear man pants and a tie. It most certainly worth investigation so let's take a look
Typical translation of this verse seem too reliant on culture. For example, skirts are not considered men's clothing in American society, generally, but in Scotland and in ancient Israel (and other places, I'm sure), men wear/wore skirts. Another factor is that if a man was wearing a skirt where it was acceptable, then went somewhere it was not acceptable, would he be violating the command? I studied the issue some and heard one person assert that "gever" might refer to soldiers rather than just generalized males. Another interesting issue is that the phrasing does not seem to be mirrored, but the two clauses are constructed with fairly different wordings (unlike the translation, which basically just reverses the first clause in the second half of the command). I think Maimonides asserted at some point that the "garment of a man is the gear of war", which seemed to relate to the other assertion I heard. One way to interpret it is that it might be a prohibition of soldiers disguising themselves as women, and women shouldn't be soldiers. This theory might be flawed, due to my lack of knowledge. The issue of women wearing/being war gear might make sense in that there were pagan ceremonies which involved such.
From my experience Torah verses usually speak in general terms, so the traditional interpretation may be correct. I did check my Interlinear Pentateuch, it seems that MT, SP and DSS are identical, so there is not so much to work with. One has to do the analysis of the text.