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All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depends on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)^n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper.