Rotating a Spherical Mirror If a plane mirror is rotated by an angle $\theta$ ( keeping the incident ray fixed) then the reflected ray gets rotated by angle $2\theta$.
Does this happen with a Spherical mirror too i.e for paraxial rays?
According to me, it will be true for paraxial rays since spherical mirrors are assumed to be flat near the pole while proving the mirror formula.
 A: If you rotate any rigid plane figure about any point in the plane through angle $\theta$, then every straight edge and tangent rotates through the same angle $\theta$. The angle between every edge or tangent and any fixed line which is not rotated, such as a ray of light, changes by the same angle $\theta$. If the ray strikes the same straight edge, or the same point on a curved edge, before and after rotation, then the reflected ray is rotated through $2\theta$, as for a plane mirror. 
The difficulty with curved mirrors (of any shape, not only spherical) is that the ray might not strike the same point Q on the mirror after the rotation. The formula works for a plane mirror because, even though the points of incidence before and after the rotation are different, the tangents at these points were parallel (co-linear, in fact) before the rotation and remain so after rotation. 

If a spherical mirror is rotated about the point Q at which the ray strikes it, then after the rotation the ray will still strike the mirror at the point Q. See case (a) in diagram above. As you note, the local surface at Q is flat and acts like a plane mirror. So rotating the mirror through angle $\theta$ rotates the reflected ray through angle $2\theta$, the same as for a plane mirror.
However, if you rotate the spherical mirror about some other point, the ray might then strike the mirror at some point other than Q, so the same relation does not necessarily continue to be true.  
For example, if you rotate the mirror about the centre of curvature C, the ray which struck the mirror at Q1 now strikes the mirror at a different point on the mirror Q2. For this symmetrical case the tangent at the point of incidence has not changed, so the reflected ray is not rotated at all. See case (b). 
If you rotate the mirror about some other point, eg the pole P, the new point of incidence Q2 is not necessarily the same as the initial point Q1. See case (c). It is possible for Q1 to be rotated to Q2, but this is a special case - the reflected ray is then rotated through $2\theta$ for the same reason as in (a). However, in most cases Q2 comes from some other point Q3, with a tangent which is not parallel to that at Q1. So the reflected ray will not in general be rotated through $2\theta$. 
