When we construct symmetry adapted linear combinations of orbitals (aka SOs) to build an MO diagram, we choose the z axis as the axis of highest symmetry and all other axis to make symmetry operations of a group easier to perform.
If we choose z-axis to point in some other direction, we will get a representation that is clearly not irreducible, but yet we would be unable to reduce it with the chosen basis because all orbitals mix (e.g. considering the case of 3d orbitals of a TM octahedral complex and having $d_{z^2}$ at a slant to the z-axis). I wonder if someone could explicitly explain why in this case the approach fails and highlight its limitations in general.
Thank you!