Let's call your coordinates $\phi$ and $\theta$, as is more usual. Thus $(x,y,z)=(sin \phi\ cos\theta, sin\phi\ sin\theta, cos\phi)$.

Yes, in the centralizer of $SO(2)$ there is the group that leaves $\theta$ unchanged, $(\phi, \theta)\mapsto (f(\phi ),\theta)$ where $f$ is a diffeomorphism from $[0,\pi]$ to itself that is nice enough at the endpoints. Also there is the group that leaves $\phi$ unchanged, $(\phi, \theta)\mapsto (\phi ,\theta +g(\phi))$ where $g$ is a smooth map from $[0,\pi]$ to $\mathbb R/2\pi \mathbb Z$. The centralizer is the semidirect product of these, and the normalizer is bigger by a factor of two.

EDIT Let's consider the related and slightly easier problem of maps $F$ from the plane to itself that commute with all of $SO(2)$, in other words maps $F:\mathbb C\to \mathbb C$ such that $F(re^{i\theta}
)=F(r)e^{i\theta}$. Such a function is determined by its restriction to $\mathbb R$. That restriction must be odd, $F(-x)=-F(x)$, by considering $\theta=\pi$. Thus if $F$ is smooth then the restriction can be written as $x\mapsto xG(x^2)$ for a smooth $G:[0,+\infty)\to\mathbb C$. Conversely, given any such smooth $G$ we can write $F(z)=zG(|z|^2)$ and get a smooth map $\mathbb C\to \mathbb C$, the unique such map commuting with rotations and having $x\mapsto xG(x^2)$ as its restriction to $\mathbb R$.

If $F$ is a diffeomorphism then $G(0)$ is different from $0$ and also $G(u)$ is different from $0$ when $u>0$. Thus $G$ can be written $G(u)=a(u)e^{ib(u)}$ where $a>0$ and $b$ are smooth real functions of $u\ge 0$. The only further constraint on $a$ or $b$ is that $x\mapsto xa(x^2)$ must be a diffeomorphism from the positive reals to itself.