Let the **cocone** of a Turing degree ${\bf d}$ be the set $cc({\bf d}):\{{\bf c}: {\bf c}\not\ge_T {\bf d}\}$. I'm curious what's known about the various partial orders (isomorphic to ones) of the form $cc({\bf d})$ for a nonzero Turing degree ${\bf d}$. Upwards cones have been extensively studied of course, and it turns out that in general the cones above distinct degrees could look very different - see e.g. [this paper by Shore](https://www.jstor.org/stable/2273376?seq=1#page_scan_tab_contents) - but I can't find much information about cocones, and they seem more finicky objects.

For example, it's not immediately clear to me how to construct a pair of degrees yielding isomorphic cocones *or* how to construct a pair of degrees yielding nonisomorphic cocones! One natural guess is that since $0'$ is definable in the Turing degrees, the cocone of $0'$ should not be isomorphic to the cocone of $0''$, since the latter has $0'$ while the former doesn't. But the definition of $0'$ I'm aware of isn't $\Sigma_1$, so conceivably $cc(0')$ could have something that looks like what $0'$ looks like in $cc(0'')$, and so on.

Basically, I'm trying to get a picture of whether, and if so to what extent, removing an upper cone from the Turing degrees could change their global structure. For example, it's [generally conjectured these days](https://math.berkeley.edu/~slaman/talks/sw.pdf) that the Turing degrees are rigid *(that is, the partial order of Turing degrees has no nontrivial automorphisms)*, but I don't see an obvious reason why this would mean that *cocones* can't have nontrivial automorphisms. My only relevant vague recollection is that the Slaman-Woodin machinery should imply that, assuming rigidity, every cocone $cc({\bf d})$ with base ${\bf d}$ *sufficiently high* will also be rigid. *(In fairness, I also have a couple sillier interests - e.g. what happens when we force with $cc({\bf d})$ (ordered by "stronger = higher"), and to what extent it depend on what ${\bf d}$ we pick?)*

To give this question focus, I'll ask specifically:

>Are there nonzero Turing degrees with nonisomorphic cocones? 

*EDIT: Note that ${\bf d}\not\in cc({\bf d})$. Were we to include it - that is, replace "$\not\ge_T$" with "$\not>_T$" in the definition of the cocone - this question would have an easy affirmative answer, since ${\bf d}$ would be definable (= the unique maximal element) and so for example the cocone of a minimal degree would be non-elementarily-equivalent to the cocone of a non-minimal degree. It's plausible to me that omitting ${\bf d}$ ultimately doesn't change much, but my current suspicion is that it changes a lot in general.*

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As a side note, there's some flexibility here with respect to how we view cocones as structures. Turing reducibility is insanely expressive: from $\le_T$ alone we can define all sorts of more complicated degree operations/relations, most importantly the jump; by contrast, in a cocone $cc({\bf d})$ it's not even clear to me that the relation "${\bf a}\vee{\bf b}\ge_T{\bf a}\vee{\bf c}$" is definable from $\le_T$ alone *(where "$\vee$" in the preceding is the join in the Turing degrees; joins in cocones won't exist in general)*. So we could conceivably get very different answers depending on what structure we provide the cocone