Can you? Yes. Answering your own questions. The trick of answering one's own question - allowing a speaker to control the "dialogue" and anticipate objections - was probably invented long before we had writing with which to record it. In the vocabulary of rhetoric, much of which dates to ancient Greece, it's called hypophora (hy-PAH-fer-uh), or sometimes anthypophora.
Is it a chaimus, or something else, when I used the following email tagline?
"How much wood would a woodchuck chuck if a woodchuck could chuck wood?
How much ground would a ground hog hog if a groundhog could hog ground?
How much pig could a whistlepig whistle if a whistlepig could whistle pig?"
Since the animal has three names, it seems a three tongue twister for each is appropriate.
Emmett
Emmett, if that's not chiasmus, it's pretty darned close, with the nice X-shape in each line.
When I think of hypophora (which I admit isn't all that often), I think of Donald Rumsfeld. I remember him doing lots of that during news conferences.
Meanwhile, I'm all aswoon at having learned there's actually going to be a movie called "Synedoche, New York."