From Ireland and Rosen's A Classical Introduction to Modern Number Theory, p.48:
Let $p$ be a prime of the form $4t+1$. Show that $a$ is a primitive root $\bmod p$ iff $-a$ is a primitive root $\bmod p$.
I can write (letting $p=4t+1$)
$\begin{align} a^{p-1} &\equiv 1 \bmod p\quad\quad \text{ because }a\text{ is a primitive root}\\ a^{4t} &\equiv 1 \bmod 4t+1\\ a^{4t} -1 &\equiv 0 \bmod 4t+1 \end{align}$
I notice that $-a$ satisfies this last equation, but I don't feel comfortable with this because I don't think this is enough to prove that $-a$ is in fact a primitive root.