The Wikipedia article "Relativistic Doppler effect - One object in circular motion around the other" confused me.
There they say that when the light source orbits the observer, the observation of the observer would be different than the case where the observer orbits the light source.
Now, what I knew from my basic physics background is that there is no such thing as "orbiting" because despite from the outside the earth seems like it is orbiting around the sun, that is just a misinterpretation. If we fix the sun at the center, the earth will seem like orbiting around the sun and if we fix the earth at the center, the sun will seem like orbiting around the earth. In essence, they both are rotating around their overall center of mass.
So, how is it possible that when I and the sun are rotating around our overall center of mass, I could observe redshift and blueshift at the same time?