Hmm, tricky one.
The intuition of the formal proof (not sure if this counts!) is that if a series converges conditionally, then it is at worst 'balanced on the edge' of convergence (root test gives 1); hence if you slightly improve convergence by decreasing $|x|$, the series converges 'properly' (root test gives something proportionally less than 1). Hence
the regions of conditional convergence are at most points balanced precariously on a boundary of a region of absolute convergence, which obviously must be symmetric.
Thus $(-1,1]$ type intervals are as asymmetric as you can get. This generalizes immediately to the complex plane, where the absolutely convergent region is trivially circular, and the conditionally convergent region is a part of the circle bounding this disc.
Morally, therefore, the symmetric nature ultimately arises from the simple fact that
the series at $+x$ converges absolutely if and only if it does at $-x$
combined with the above observation that conditionally convergent regions are 'narrow', 'unstable', 'borderline cases', etc.