Is Blum's theorem for acceptable enumerations of partially computable functions bidirectional?
For the standard enumeration $\varphi$ and some computable permutation $h$,
$$\forall e;~ \psi_e = h^{-1} \varphi_{h(e)}h ~~~~(*)$$

We want to prove that such $\psi$ is an acceptable enumeration. It suffices to find computable functions $f$ and $g$ such that for all $e$, $\psi_e = \varphi_{f(e)}$ and $\varphi_e = \psi_{g(e)}$.

$h^{-1} \varphi_{h(e)}h$ is partial computable. So by s-m-n theorem, there is a computable function $f$ such that for all $e$, $h^{-1} \varphi_{h(e)}h = \varphi_{f(e)}$, therefor by $*$ we have, $\psi_e = \varphi_{f(e)}$

$h \varphi_e h^{-1}$ is also partial computable. So again by s-m-n theorem, there is computable $g'$ such that for all $e$, $h \varphi_e h^{-1} = \varphi_{g'(e)}$. Define computable function $g$ as $g = h^{-1} g'$. In $*$ replace $e$ with $g(e)$ to get 
$$\psi_{g(e)} = h^{-1} \varphi_{hg(e)}h ~.$$
By definition of $g$, we have $h g = h h^{-1} g' = g'$, so
$$\psi_{g(e)} = h^{-1} \varphi_{g'(e)}h ~.$$
By definition of $g'$
$$\psi_{g(e)} = h^{-1} h \varphi_e h^{-1} h ~,$$
and thus
$$\psi_{g(e)} = \varphi_e ~. ~~\Box$$