What is the normalizer of the circle in the diffeomorphism group of the 2-sphere?  What is the normalizer of $SO(2)$ in $\mathrm{Diff}(S^2)$?
Remarks:


*

*We let $SO(2)$ act on $S^2$ via the rotation about the $z$-axis. 
It is immediate that each element of the normalizer must map any parallel 
(i.e. a fiber of the projection of $S^2$ onto the $z$-axis) 
to a parallel. 

*Of course, the normalizer
contains $O(2)$, as well as the following elements that ``push along the meridians''. Let $(\phi, r)\in [0,2\pi]\times [0, \pi]$ be coordinates on $S^2$
with parallels given by $r=\mathrm{const}$. Define a self-map of $S^2$
by $H(\phi, r)=(\phi, r+h(r))$ where $h$ is some smooth function subject
to the boundary conditions ensuring that $H$ is a diffeomorphism
(e.g. $h$ vanishes near $0$ or $\pi$). Then $H$ commutes with the
$SO(2)$-action, that is given by translation in $\phi$. 
 A: 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. 
