Neither nor rotate about the other

Neither nor rotate about the other.

Both the Sun and the Earth orbit around the centre of mass of the solar system which is known as the solar system barycentre.

The position of the barycentre is constantly changing as the relative positions of change. The barycentre can be anywhere from close to the centre of the Sun to about 1 million kilometres from the centre of the Sun.

The diagram shows how the position of the solar system barycentre changes over the period 1938-2052. The units on the axes are solar diameters 696,300km.