Riemann Zeta Function:
$$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\frac{1}{5^{s}}+\frac{1}{6^{s}}+\frac{1}{7^{s}}+\frac{1}{8^{s}}+\frac{1}{9^{s}}+\frac{1}{10^{s}}+\dots$$
"Reverse $2$" Riemann Zeta Function:
$$\frac{1}{1^{s}}+\color{red}{\frac{1}{\sqrt2^{s}}}+\frac{1}{3^{s}}+\color{red}{\frac{1}{2^{s}}}+\frac{1}{5^{s}}+\color{red}{\frac{1}{\sqrt{18}^{s}}}+\frac{1}{7^{s}}+\color{red}{\frac{1}{\sqrt{8}^{s}}}+\frac{1}{9^{s}}+\color{red}{\frac{1}{\sqrt{50}^{s}}}+\dots$$
"Reverse $3$" Riemann Zeta Function:
$$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\color{blue}{\frac{1}{\sqrt3^{s}}}+\frac{1}{4^{s}}+\color{blue}{\frac{1}{\sqrt{35}^{s}}}+\color{blue}{\frac{1}{\sqrt{12}^{s}}}+\color{blue}{\frac{1}{\sqrt{35}^{s}}}+\frac{1}{8^{s}}+\color{blue}{\frac{1}{3^{s}}}+\frac{1}{10^{s}}+\dots$$
"Reverse $10$" Riemann Zeta Function:
$$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\frac{1}{5^{s}}+\frac{1}{6^{s}}+\frac{1}{7^{s}}+\frac{1}{8^{s}}+\frac{1}{9^{s}}+\color{green}{\frac{1}{\sqrt{10}^{s}}}+\dots$$
In general, a Reverse $x$ Riemann zeta function takes $n=1,2,3\dots$ and multiplies it with its reverse value (reverse digits) in number base $x$, then takes the square root of the product. After that, raise it to the power of $-s$ as you would for zeta(s).
As you can see, if $n$ is a palindrome in number base $x$, then it remains $n$. Otherwise, it becomes the $\sqrt{n\times\overline{n}}$ where $\overline{n}$ is the number $n$, but its digits are reversed when written in number base $x$.
The value of "Reverse Zeta $x$" of $s$ tends to $\zeta{(s)}$ as $x$ grows bigger. Suppose we take the limit $x\to\infty$, then we have a normal zeta function. It's because that makes the number base a base where each number is a single symbol (one digit), meaning all $n$s reverse back to $n$.
Does something on this already exist somewhere or is this the first time this is mentioned?
How can we check when this will converge for a given $x$?
I suppose the bound slightly varies based on $x$ and converges to that the real part of $s$ must be $\gt1$ as $x$ tends to $\infty$?Can one find closed forms for some $x$ and some $s$?
For example, what is the (is there a) closed form of this when $x=2$ and $s=2$ ?
Can we express the "reverse $x$ zeta function" in the terms of the zeta function?
Also, I couldn't find anything on the reciprocal sums of palindromic numbers.
Note: Not a complete answer, but here is some Mathematica code that can evaluate this sum and is good to play around with the behavior of your sum.
It's most likely not optimized, since I'm fairly new to Mathematica, but nonetheless. This will generate a table of plots of $\zeta_b(s)$ from $s = 1 \dots 5$ and $b_{base} = 2 \dots 4$, where $\zeta_b$ denotes the reverse base zeta with base $b$. To change some of the parameters: In the second line
{s, 1, 5}is{s, <begin_index>, <end_index>}so you can change that to change the increment of $s$. Same goes with the base denoted as $b$. There's a free online Wolfram Lab you can use to test this if you don't have Mathematica. Below is a table of plots generated by this code, where the $x$-axis serves as the bound of the summation: