The first applet below shows the Euler-angle parameterization of the rotation group, with $r$ controlling the size of the cube, $\alpha$ and $\gamma$ denoting rotations about the $z$ axis, and $\beta$ a rotation about the $x$ axis. The rotations are performed in the order $\alpha$, $\beta$, $\gamma$. You can think of ($\beta$,$\gamma$) as determining the (co)latitude and longitude to which you move the north pole, and $\alpha$ as the angle of rotation about this resulting axis.
Use the sliders to rotate the cube(s).
(These applets were created with
GeoGebra.)
The second applet shows rotations of the cube around axes through corners (green; $\eta$) and edges (blue; $\xi$).