Is rotation of a mirror image allowed in order to determine chirality? Doesn't rotation alter the mirror image of the molecule?
When checking out chirality or achirality you are always allowed and supposed to rotate the molecule freely in all dimensions. Essentially, what you are trying to prove, is that the mirror image of a molecule can be achieved by rotation alone.

The thought behind this is simple. Chirality is macroscopically proven by optical activity. If you reduce the experiment to its bare basics, you have a polarised light beam that passes through a solution and hits a single molecule to be deflected either to the right (dextrorotatory) or to the left (levorotatory). A second molecule it hits which has the same chirality will move it in the same direction by the same amount. If, however, it hits a molecule which is exactly the mirror image of the first, the effect will be cancelled and the plane of polarisation is moved back into the vertical. In a solution containing only one enantiomer of a chiral substance, the beam will never hit a mirror image and thus always be deflected in a certain direction.

If we continue this thought experiment, we realise that molecular motion in solution is random. Thus, a molecule can — if we give it conciousness — move around freely and let it try to assume exactly the mirror image of the previous one. Therefore, we are allowed and encouraged to rotate (but not to use another mirror image or inversion!) our molecule until it ‘fits’.

By the way, most if not all achiral compounds can also be rotated before you apply the plane of symmetry. Oftentimes if not always, you can rotate them in a way that you see the plane of symmetry present in the molecule itself. If you managed to do that, you won, because just use a plane parallel to that one for mirror imaging and the image will automatically be identical to its template without further rotation.

The plane of symmetry in isopropanol cuts through the $\ce{CHOH}$ bit of isopropanol, dissecting each of those four atoms into two half-spheres. One methyl group will be transformed exactly onto the other by this plane.