IN the time of Moses there was a religious sect that as part of their rituals both sexes took on looks and behaviors of the other sex. Along with this, in the tribes under Moses, there remained the customs of the Hebrews. The Hebrews were sexually segregated. What would happen is WOMEN would dress like men so they could go over to the male areas and have illicit sex (I am sure some men did the opposite). Then there were the men that dressed as women to get out of fighting.
As can see there were many reason for deut 22:5's writing. But there were many other laws in Deut that NO ONE pays mind to these days, and many have the word "abomination" in them.
Every educated discussion I have read or had concerning this verse says it has NOTHING to do with modern day crossdressing. It had to do with infidelity, adultry, sex outside of marriage, and cowardice. If people actually followed the old testement there would be MANY MANY dead people. Newt Gingritch, Donald Trump, Madonna, to name a few would be dead for their adulterous actions. I would be dead for mouthing off at my parents when I was younger.
Thank the maker we do not follow that old book.
