I understand that the CMB rest frame for a typical FLRW universe should coincide with co-moving coordinates, but under what conditions won't the two coincide?
For example if the universe had some net spin (an abhorrent thought I know!) then would the two frames still coincide?
EDIT: I need to explain further. Say the Universe is the closed FLRW model, EXCEPT that it has a net spin. In this case the rest frame of the CMB would necessarily be different from comoving coordinates. Wouldn't the frame in which space appeared isotropic and homogeneous not be comoving but rather moving with center of momentum frame? If the three-sphere's large enough, there appears no deviation from isotropy within an observer's hubble radius. Of course there's a curious situation at the "poles" of the rotation where the isotropic frame is a rotating one (making me think towards Mach's principle). anyway, this was my attempt at a an example for my question. thanks