Ah. The texture was nothing sophisticated; I just opened up a basic one in gimp, and erased a chunk to transparent, to see what would happen.
The texture I was seeing "behind" was presumably the stuff remaining in the other three (i.e. non-alpha) channels.
Re: alpha=specularity, that could be fun to play with, too. How does that work? Presumably one extreme of the alpha channel maps to very "shiny", and the other extreme maps to matt/non-reflective?