In SR, only inertial reference frames are equivalent, and so you can't tell which object moves only by experiments performed in those different reference frames. However this is not the case for non-inertial reference frames, which moves with acceleration. Earth rotates around sun, thus it has centripetal and centrifugal pseudo force acting on it, so this makes Earth a non-inertial reference frame cause now it has centripetal acceleration, which in principle can be measured. Common form of gravitational acceleration on Earth surface is :
$$ g= G\frac{M}{r^2}-\left(\frac{v_{e\perp}^2}{r}~\pm\frac{v_{s\perp}^2}{R}\right) $$
Where $M$ is Earth mass, $r$ is Earth radius, $R$ is Earth orbit radius, $v_{e\perp}$ - Earth rotation around own axis tangential speed and $v_{s\perp}$ is Earth rotation around Sun tangential speed. Sign $+$ or $-$ in formula depends on where are you are measuring gravitational acceleration - in Earth side directed towards sun, or in the side directed outwards from the sun (then Earth axis rotational bulge effect adds to the Sun centrifugal force).
Centrifugal acceleration due to Earth rotation around own axis is small, about $0.03 ~m/s^2$. Centrifugal acceleration due to Earth orbiting Sun is even smaller, about $0.005 ~m/s^2$. But it's not zero and can be measured as variations in gravitation acceleration over time period of day (dependently on your relative radial position towards the Sun). If like you said not Earth would be rotating around Sun, but Sun instead around us - we would not get such third term in equation and would not experience mentioned gravity variations in time. Thus Earth/Sun non-inertial reference frames are not identical to each other, and which is which- can be measured by local experiments in RF.