"A woman must not wear men's clothing, nor a man wear women's clothing, for the Lord, your God, detests anyone who does this" (NIV)
"The woman shall not wear that which pertaineth unto a man, neither shall a man put on a woman's garment: for all that do are an abomination unto the Lord" (KJV)
Since at that time men and women dressed essentially the same - there really wasn't "men's only" or "women's only" clothing - then it wouldn't seem to be talking about "cross"-dressing. The research on the meaning of this verse has yielded many different and quite plausible suggestions.
One suggestion centers on deception for wrongful purposes. Proposed by Rev. Hotchkiss, it is her estimation that because women were prohibited from entering into the Temple, the first part addresses women trying to sneak in disguised as men. And it is her opinion that, because men and women were usually housed separately within a family or tribe, the second phrase is intended to keep men from being able to sneak into the women's tents or homes. Thus, deception...not personal identity...is the issue.
Other theories offered indicate that this passage is directed at those who would wear the armor and shield of a female warrior goddess in the worship of her. One consistent message from God is to not worship others. Our love for God must be true. Another suggestion is that it is aimed at ending certain ancient fertility rites or sexual practices commonplace in the day. Yet another possibility was that, according to some, there were enemies of Israel who would disguise themselves as women to infiltrate the land. The protection and purity of the tribal peoples was paramount. None of these ideas hint at gender transformation.
Another view factors in the cultural aspects of the verse. As I mentioned, men and women essentially dressed the same, had the same style of clothing, when the verse was written. In more "modern" time, pants as we know them, didn't come into existence for a thousand years after Christ. This verse was from several thousand years before Christ. And yet, it was this verse that Joan of Arc was ultimately convicted. She wore pants and refused to remove them while she was incarcerated in a men's jail as she was fearful of rape. (At the time, the practice was to house women inmates at nunneries). And so she was burned at the stake. And now she is a Saint in the eyes of the very same church that convicted her. Which is it? Were they "right" then? Or are they "right" now?
Scottish men wear kilts. And yet some non-Scottish Christians deem kilts to be the equivalent as skirts for women. While similar in design, the kilt and the skirt were developed independently of each other and with different cultural motives. Scotsmen are often targeted with this verse for their "abomination". Again, given when the verse was authored, it could not have been talking about Scottish kilts.
And in today's culture, it is quite acceptable for women to wear jeans, t-shirts, shoes or boots that were all initially designed for men. Only a few, extremely conservative denominations have any prohibitions to women wearing such clothing. Indeed, it is quite acceptable. And yet the verse addresses both men and women. You'd think what's good for the goose, would be good for the gander (or something like that!) in current usage. Let a girl go to school wearing sneakers, jeans and a t-shirt and no one says a thing - not even the Christians among the staff. Let a boy go to school wearing a skirt and all heck breaks loose - even among the non-Christians! Either this passage applies or it doesn't.
Some would argue that, regardless of the time and place of its authorship, it still is what the Bible says. Then why are we not beholden to many of the surrounding verses of Deuteronomy? Those texts include not yoking an ox and an ass together, not wearing clothing of mixed linens (imagine those t-shirts of cotton/poly blends!), not rotating crops (a disastrous notion if you are a farmer) and requiring the stoning of non-virgins. Again, either these also apply to the present day.....or they don't.
And even after all of that - this verse does not talk about someone changing from one gender to the either. It's about appearance, not identity.