WOW... any chance someone is playing a tick on you?

I have never seen anything like that on any 60 to 66 trucks I have been around. I don't think it could be dash to windshield trim because that would be putting the screws into the rubber.
Maybe it goes to a panel wagon if it has a head liner or something.. I am not the panel guy.
To be honest it looks like a head liner trim piece on a 58 Pontiac.. I remember seeing something similar there...