Are any points at rest on a globe spinning both horizontally and vertically? Suppose there was a globe which could be spun about polar and horizontal axis. It is first spun about the polar axis, then promptly spun about the horizontal axis. Would any points be at rest?
 A: Yes. There always has to be an axis for the rotation and therefore there are two points that are at rest. This is sometimes illustrated with the comparison to a football during a match, which is put at the kick-off point right before the first and second half-time of the game and turned a lot in many different directions in between. There are always two (or infinitly many points if you have a lucky day) that are in the exact same position both times:

Mathematically speaking, the rotation is described by a matrix $A\in\operatorname{SO}(3)$ (third special orthogonal group or also called rotation group, see here), which always has an eigenvector as its characteristic polynomial $\chi_A(X)=\det(X1_3-A)$ is a polynomial of third order and therefore always has a real root. You find more under Euler's rotation theorem. The source of the image is the corresponding german page.
A: the components of a  point on sphere surface are:
\begin{align*}
&\mathbf{R}=\begin{bmatrix}
              x \\
              y \\
              z \\
            \end{bmatrix}=
\left[ \begin {array}{c} \cos \left( \theta_0 \right) \sin \left( \phi_0
 \right) \\ \sin \left( \theta_0 \right) \sin \left(
\phi_0 \right) \\ \cos \left( \phi_0 \right)
\end {array} \right]
\end{align*}
you rotate the  position vector $~\mathbf R~$  to obtain the final position vector $~\mathbf R_f~$
\begin{align*}
 &\mathbf{R}_f=\mathbf S_y(\varphi_2)\,\mathbf S_z(\varphi_1)\,\mathbf{R}=
\left[ \begin {array}{c} \cos \left( \varphi _{{2}} \right) \cos
 \left( \varphi _{{1}} \right) x-\cos \left( \varphi _{{2}} \right) 
\sin \left( \varphi _{{1}} \right) y+\sin \left( \varphi _{{2}}
 \right) z\\  \sin \left( \varphi _{{1}} \right) x+
\cos \left( \varphi _{{1}} \right) y\\  -\sin \left( 
\varphi _{{2}} \right) \cos \left( \varphi _{{1}} \right) x+\sin
 \left( \varphi _{{2}} \right) \sin \left( \varphi _{{1}} \right) y+
\cos \left( \varphi _{{2}} \right) z\end {array} \right] 
\tag 1\\
\end{align*}
where $~\varphi_1~$ the rotation about the z-axes and
$~\varphi_2~$ the rotation about the new y-axes .
from here e.g. with the request that $~\mathbf R_f=[0,0,1]~$ you obtain from equation (1) that $~\varphi_1=-\theta_0~,\varphi_2=-\phi_0$
