I really don't know how to classify this weave. It's based on the 3-shaft twill (jeans twill) and it resembles crackle, but it has its own logic. First, the blocks. Each block is threaded as a 3-shaft twill on 3 of the 4 shafts of a 4-shaft loom. This gives you four possible blocks in the threading: 123, 234, 341, and 412. I've altered the order somewhat to make the transitions from one block to another smoother. So what I have come up with are: Block A, 231; Block B, 234; Block C, 134; and Block D, 124.
Transitions are smooth if you go clockwise in the block rotation (ABCDABCD, etc.) or counterclockwise (ADCBADCB, etc.) You may not, however, jump to opposite sides of the circle without special adjustments to the threading transitions.
Use the standard 2/2 twill tie-up and treadle as drawn in. This will give you four blocks of a hybrid twill-like structure: not a twill, not a plain weave, but a crepe-like combination of the two, each block emphasizing the threads on a particular block.
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Here is my first dive into the weaving, with a three-color warp rotation and a one-shuttle weft.
A two-color warp rotation will give other results. There is much opportunity for play here. There are a couple of examples in my previous post.
There is no true tabby but the usual 13, 24 treadling will give an acceptable basketweave for hems.
What endears this structure to me is its ease of warping! You hold three ends in hand in a the threading cross and the raddle cross, then pick the colors in strict rotation as you thread the heddles. The loom prep goes quickly, and the counting is easy. This is a great opportunity for using up partial cones for stash-busting.
And it's a one-shuttle weave! What's not to love?
I don't remember having seen this structure described in the literature, but I'm sure someone somewhere has thought of it. If you know of some references, I'd love to hear from you.
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Now, in the Department of Books, I have just finished reading Grammar for a Full Life: How the Ways We Shape a Sentence Can Limit or Enlarge Us by Lawrence Weinstein. Highly recommended.