Suppose we have $n$ samples $s\_1,...,s\_n$ from an unknown real-valued distribution $D$. We are interested in a statistic to test if $D$ is symmetric around zero. (In my application, $n$ is only about 50, so I'm interested in the non-asymptotic regime.)

If $D$ is symmetric and $s\sim D$, then $\Pr(s>0)=\Pr(s<0)$, so by comparing the count of positive versus negative $s\_i$, we get a simple statistic for a zero median, which is necessary for symmetry. In the hopes of a stronger test, I was going to use a two-sample Kolmogorov-Smirnov test to compare $s\_1,...,s\_n$ with $-s\_1,...,-s\_n$. But then it belatedly struck me that perhaps I wasn't the first person to think about this problem :)

Any reference suggestions or (ideally) actual test statistics would be very much appreciated.