On Wed, 20 Jan 2016 12:42:43 -0500, Ralph Mowery wrote:

One other thing that should not need to be said. When you flip it back on,
did you turn it all the way to OFF before turning it back on ?

Good suggestion, but, yes, I instinctively did that, but, I just
checked again, and it still flips immediately to the trip position.

Here's a picture of it off.
http://i65.tinypic.com/i5692c.jpg

Here's a picture of it in the trip position.
http://i66.tinypic.com/2ec2kol.jpg

It's the breaker labeled "R" on the right (4 breakers down from the top).
15 amps. Seems to be "taped?" together with another one, which is
independent.

(Why the tape?)
Maybe it's a non-ganged dual unit?