For an object, any unique centre and, more generally, any point with unique properties with respect to the object is a fixed point of its symmetry group.
In the case of a physical body, if for the symmetry not only the shape but also the density is taken into account, it applies to the centre of mass.
If the set of fixed points of the symmetry group of an object is a singleton then the object has a specific centre of symmetry.
The centroid and centre of mass, if defined, are this point.
If the set of fixed points of the symmetry group of an object is a line or plane then the centroid and centre of mass of the object, if defined, and any other point that has unique properties with respect to the object, are on this line or plane.