⋇ "Division Times" operator in Unicode (U+22C7)?

The intent ought to be what you said. So, it is $a ⋇ b$ is short for "$a \ b$ or $a \ b^{-1}$" that is the multiplicative analogue of $\pm$.

Yet, a more common way to write $a ⋇ b$ is just $a \ b^{\pm 1}$.

In addition, I never heard of this operator, all the early search hits are for Unicode-tables not math, and at least to me it looks quite similar to an asterisks at small font size.

In short, it seems more like an artifact of getting a symbol-set somehow 'complete.' Rather than something that is actually used.

That said, I just learned that there is a LaTeX (needs amssymb) and also MatjJax command for it, it is \divideontimes giving $\divideontimes$

No, it doesn't behave like $\pm$. An important part of how $\pm$ behaves is that it is likely to be understood without further explanation, and the $⋇$ sign does not have that property.

You can use it for "divided or multiplied by" if you want, but you will have to explain this usage to your reader before you do so, because you can't expect it to be clear that is what you want.

I would expect it is most likely to be used when someone needs a symbol for a new thing that behaves syntactically like a multiplication-like operator but is not actual multiplication -- and there is reason not to use the more common choices such as $*$, $\star$, $\otimes$, $\circledast$, and so forth.