I was exploring the layout of primitive roots of primes over a reasonable range and this question concerns the number of primitive roots either side of $p/2$.
Many primes have an exact match between what I call lower and upper primitive roots, those below and above $p/2$. For the most part, these are the primes $p\equiv 1 \bmod 4$ - these have primitive roots that are symmetric, $x$ is a primitive root meaning that $-x$ is also a primitive root.
However my question really concerns the primes $p\equiv 3 \bmod 4$. For these primes, the primitive roots are antisymmetric: $x$ being a primitive root implies that $-x$ is not a primitive root. In these cases there is some scatter but there is also a definite trend to have more primitive roots in the upper category. I can understand that there are more natural squares in $(0,p/2)$ than in $(p/2,p)$ but the imbalance trend seems to be slightly larger than that value as shown here in a plot of the upper root count minus lower root count:
Does anyone have some insight into what drives this imbalance in primitive roots, favoring higher values over lower?