Reordering a divergent series can indeed give you any result you want. This is not reordering though, it’s called Cesàro summation. It won’t always get you an answer, but if it does then the answer is unique and reacts nicely to sums and products.

Talking about it as summation might be misleading since that’s one of those concrete terms that mathematicians like to redefine without anyone’s approval. Picture we have a library that includes many tricks and approaches for taking an infinite series as input and outputs a number. We know it works as expected on every convergent series. But we forgot to put in any preconditions and we’ve let people input things that are not convergent series. But whoa in many cases we are still getting a number out of it, and it’s always the same answer no matter what we do. Maybe that’s something worth studying?

> This is not reordering though, it’s called Cesàro summation. It won’t always get you an answer, but if it does then the answer is unique and reacts nicely to sums and products.

It's also probably worth noting that, if the series is genuinely summable, then Cesàro summation gives its sum.