Why does acceleration break symmetry in special relativity?

As I understand it, in special relativity, when two bodies are moving away from one another, there is no absolute sense in which one is moving away from the other (e.g. when you jump in the air it is equally true to say that the Earth is moving away from you as it is to say that you are moving away from the Earth; or when a spaceship is going deep into space it's just as true to say that the Earth is moving away from the spaceship as it is to say the spaceship is moving away from Earth).

This has interesting implications when it comes to the more funky aspects of special relativity (i.e. time dilation, length contraction). Because this means that if Bob is moving close to light speed relative to Jane, Bob will perceive Jane as experiencing length contraction and time dilation, but Jane will not experience these things. From her point of view, it is Bob that is experiencing length contraction and time dilation. So both will always experience the other as experiencing these things, because from their point of view it is always the other person moving at near light speeds. So special relativity is symmetrical this way.

As I understand it though, this symmetry breaks when it comes to acceleration. This is how you can have a scenario where e.g. Bob ages a lot compared to Jane (because he accelerated or decelerated more).

So my question is: why does this symmetry in special relativity break when it comes to acceleration?