Working from right to left.

[bo198214]

Well other than the facts that x^^0 should be x√x and (x^^y)√(x^^z) should be x^^(z-y)

Something that "should be" can not be a fact
And it is not satisfiable.
It is not even true that (x^^y)^(x^^z) = x^^(y+z).
One can prove that any suitable operation ^^ which satisfies this equation is trivial, e.g. x^^y=1.
Neither left-to-right tetration nor right-to-left tetration nor symmetric/balanced tetration satisfies this anticipated property.