The standard rearrangement theorem for conditionally convergent series says that the terms in a conditionally convergent series can be rearranged so that the new sum is any desired number, or $\pm\infty$.

Let me set up some notation to investigate this a little further. My coefficients will be determined by a function $\alpha:\mathbb{N}\to\mathbb{R}$ (or $\mathbb{C}$). Then I'll simply write $\sum\alpha$ for this series $\sum_{n=1}^\infty \alpha(n)$.

Now we let $\operatorname{Sym}(\mathbb{N})$ act on the set $A = \{ \alpha: \mathbb{N}\to \mathbb{R} \}$ by the rule $\alpha^\sigma (x) = \alpha(\sigma(x))$. Then we can ask simple questions

1. is $\sum \alpha^\sigma = \sum \alpha$?
2. if $\sum\alpha$ converges, does $\sum\alpha^\sigma$ also converge?

But these are not really the questions I want to ask.

Let $B\subseteq A$ be the subset of all sequences $\alpha$ so that the series $\sum\alpha$ is conditionally convergent.

Question 1 What is the stabilizer $G = \{ \sigma \mid \alpha^\sigma \in B\ \forall \alpha\in B\}$?

Question 2 What is the group of sum-preserving permutations $H = \{ \sigma \mid \sum \alpha^\sigma = \sum\alpha \ \forall \alpha\in B\}$?

Certainly any permutation which is the identity off a finite subset of $\mathbb{N}$ is contained in both.

There are some subgroups of $\operatorname{Sym}(\mathbb{N})$

This is related to the question here.