This is an excercise in the book Algebraic Topology - Greenberg and Harper
Excercise : Let $f$ and $g$ be a map from $S^n$ to $S^n$ such that $f(x)\neq g(x)$ for all $x$
Then $f $ is homotopic to $ag$ where $a$ is antipodal map, hence deg($f)=(-1)^{n+1} $deg($g$).
Here, deg($f$) is a map from $H_n(S^n; {\bf Z})$ to $H_n(S^n; {\bf Z})$ induced from $f$
Question : If $n=2$, and if $f$ and $g$ are rotations with different axes, then $f(x)\neq g(x)$ for all $x$. So deg($f)=$deg($g$)$=1$ and deg$(a) =-1$. So is the exercise right ?