How to align two ellipsoids along center and long axes?

I’m struggling to understand rotation/translation transforms as applied to things like spheres and ellipsoids. Take the following silly example.

Given this pair of random ellipsoids, how do I find a rotation/translation that aligns their centers and their long axes? (no matter that they differ in sizes, only care about their centers and long axes being aligned). Are the two possible geometric transformations equivalent? (Say, moving ellipsoid 1 to be aligned to the fixed ellipsoid 2, and vice versa?)

SeedRandom[1234]; twoRandomEllipsoids =    Table[{Ellipsoid[RandomReal[10, {3}], RandomReal[{1, 2}, 3]]}, {2}]; Graphics3D[{Opacity[0.2], twoRandomEllipsoids}, Axes -> True,   AxesLabel -> {x, y, z}] 

enter image description here

Thanks!