Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$:
"for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$."
Say that $A$ and $B$ are *threadable* (my terminology) iff
there is a scaling and rotation of $B$ to $B'$ such that
(a) Every vertex of $A$ is exterior to $B'$, and
(b) Every vertex of $B'$ is exterior to $A$.
(I am exploring this notion of "threadability" as a measure of shape similarity.)

Two examples.
**(1)** For $A$ a cube (blue) and $B$ a cuboctahedron (red), $(A,B)$ is threadable, e.g.:

Note that $A$ and $B$ are not duals; for duals, threadability is obvious.

**(2)** For $A$ a truncated icosahedron (red) and $B$
a vertically stretched pentagonal bipyramid (blue),
I believe (but have not proved)
it is not possible to scale & rotate $B$ to *thread* with $A$:

Computing whether or not $A$ and $B$ are threadable seems quite difficult, only achievable exactly via an $O(n^k)$ algoithm for $n$-vertex polyhedra, for $k$ an exponent that captures all the combinatorial possibilities. Perhaps $k=6$ would be necessary; I haven't thought that through carefully, but certainly it would a high computational complexity.

So, here, finally, is my question.

Q. Are there succinct sufficiency criteria for when a pair $(A,B)$ are guaranteed to be threadable?

What I have in mind here is something like this:
"*If the diameter/width ratio of $A$ and $B$ is approximately (or even: exactly) the same,
then $A$ and $B$ are threadable.*"
I don't believe this, but it gives the flavor of sufficiency conditions I seek.
I have a sense that no such "simple" sufficiency conditions exist, because of the
seeming dependence upon the micro- combinatorial structure of $A$ and $B$.
But perhaps others can see more clearly through this thicket than I ... ?