Let $A: S^n \to S^n$ denote the map sending a point on a sphere to the exact opposite point. Let $A_*: H_n(S^n) \to H_n(S^n)$ denote the action of $A$ on the homology group. Then $A_*$ is an isomorphism from $\mathbb{Z}$ to $\mathbb{Z}$, but the generator might change.
For instance, on $S^1$, if $x_0$ is the base point on a clockwise loop and $x_1$ is a point a little clockwise of $x_0$, then their image under $A$ will have the same orientation, so $1$ will map to $1$ by $A_*$. On $S^2$, however, a clockwise loop on the top of the sphere will map to a counterclockwise loop on the bottom, so here $1$ maps to $-1$.
What can I say about higher dimensional spheres?