Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}$ is at least $C$ ($f(\mathcal{R}x)$ is a rotation of $f$ on $S^2$). Does there exist $I\neq\mathcal{R}\_0 \in SO(3)$ such that $f(x)=f(\mathcal{R}\_0x)$ on an open subset of $S^2$? 



Symmetry of functions on $S^2$
Note that any eigen-function satisfies your condition for some $C>0$.
It remains to find a non-symmetric one. Take for example 
$$f(x,y,z)=10\cdot x^3+y^3+\tfrac1{10}\cdot z^3.$$


