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?

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 $~\varphi_1=-\theta_0~,\varphi_2=-\phi_0$
