As pointed out in the comments I have realized that I am taking a Mellin transform of a function $\psi(x)$ defined by:
$$\psi : (0,\infty) \to \mathbb{C}$$
$$\mathcal{M}(\psi)(s) = \int_{0}^{\infty} \psi(x) x^s \, \frac{dx}{x}$$
for $s \in \mathbb{C}.$ And it seems what I've done is swapped the complex $s$ for a real $s$, repeating the process with this real $s$:
$$\mathcal{M}(\psi)(s):(0,\infty)\to \Bbb C.$$
Mellin transform pairs are ubiquitous in analytic number theory. Maybe the most famous pair is $e^{-x}$ and $\Gamma(s).$ Define a "Mellin triad" (which are quite rare as pointed out in the comments) by the discrete pre-periodic forward orbit of Mellin transforms of a function $\psi$:
$$O_\mathcal{M}(\psi)=\bigg\lbrace\psi, \mathcal{M}^{(1)}(\psi),\mathcal{M}^{(2)}(\psi) \bigg\rbrace$$
Clearly $$\bigg\lbrace \psi,\mathcal{M}^{(1)}(\psi)\bigg\rbrace, \bigg\lbrace \mathcal{M}^{(1)}(\psi),\mathcal{M}^{(2)}(\psi)\bigg\rbrace$$
are our Mellin pairs. How do we understand:
$$\bigg\lbrace \psi,\mathcal{M}^{(2)}(\psi)\bigg\rbrace?$$
It's not a Mellin pair because the functions are seperated by two Mellin transforms as opposed to just one.
Now I know that an integral transform generally maps a function to another function space where one might be able to leverage nicer properties of the new function and solve the problem and then transform the solution back to the original domain.
But does this hold for two functions that are twice-separated by Mellin transforms or does it just hold for Mellin pairs?
I guess I want to understand exactly how much information is preserved when passing from $\psi \to \mathcal{M}^{(2)}(\psi).$
For two explicit examples of Mellin triads, define:
$$O_\mathcal{M}(\psi)=\bigg\lbrace e^{\frac{1}{\log x}}, \frac{2K_1(2\sqrt{z})}{\sqrt{z}}, \Gamma(s)\Gamma(s-1) \bigg\rbrace$$
$$O_\mathcal{M}(\psi)=\bigg\lbrace\sum_{n=1}^\infty e^{\frac{nt}{\log x}},\sum_{n=1}^\infty 2\sqrt{nt}K_1(2\sqrt{nt}),\zeta(s)\Gamma(s)\Gamma(s+1)\bigg\rbrace$$
Where $K_1$ is a modified Bessel function of the second kind, $\zeta(s)$ is the Riemann zeta function, and $\Gamma(s)$ is the Gamma function.