EDIT 5/13/09: The painting "In the Midst of Joy, We Cry" is now for sale here.
The form was inspired by several other repeating/interlocking forms, and it works like this:
The two refrains, marked "A" and "B," set the rhymes. Other lines rhyming with them are marked "a" and "b." The verses are quatrains, always with a 3:1 ratio between the two rhymes, but the arrangement sequence varies. Both refrains begin the first verse; A begins the second verse; B begins the third verse; and both conclude the final verse but in reverse order (BA). The tight rhyme scheme thus contrasts with the shifting structure.
I couldn't think of a name for the form, so I posed that as a question to my donors. The winning name was kentron, submitted by minor_architect. She explains:
I'm leaning toward the Greek word kentron (κέντρον) or some variant thereof. It's the basis for the Latin word centrum, "center." This seems like a good description of your poetic form since it stands in the center of the three other forms you mentioned (Mirrored Refrain, Rondel, Villanelle). Plus, it's a nice play on the title of the first poem you used it for!