I started to wonder if there exist a continuous map $f \colon S^n \to S^n$ such that $\mathrm{deg} f = 2$ and $f$ is even, i.e. $f(x) = f(-x)$. This question is only interesting for odd $n$ since if $n$ is even then every even map $f$ has $\mathrm{deg} f = 0$.
It was a homework exercise to prove that every even map has even degree, and I started to wonder if all even degrees are realized by even maps.
I was thinking that maybe the mapping $g(x,t) = \begin{cases} 2 \sqrt{\frac{t}{t-1}}x, 1 + 2t, &\text{if } -1 \le t \le 0 \\ -2 \sqrt{\frac{t}{t+1}}x, 1 - 2t, & \text{if } 0 \le t \le 1\end{cases}$ would work. (Here I have separated the last coordinate in $\mathbb{R}^{n+1}$.) It maps the lower hemisphere to the whole sphere and does the same for the upper hemisphere in reverse order. I have no idea how to prove anything relevant about this mapping, though.