I find it interesting that we skip over the word "wear" in
Deut 22:5. That definition is used only once in the OT - only once. It is not the normal definition for putting on clothing.
Wear:
H1961
היה
hâyâh
haw-yaw'
A primitive root (compare H1933);
to exist, that is, be or become, come to pass.
I believe this verse goes a lot deeper than simply donning a man's/woman's apparel.