Title kinda says it all.
The most frequent use that I have for Move/Copy Body (M/CB) feature in a component is when I receive a part from manufacturer which is not oriented upon the origin and major planes as I wish to have it useful for my designs. Most commonly among these cases is for downloaded control panel enclosures, where the model provided clearly represents their process of assembling the product while it is lying on its back, which is its mounting surface. I intend to mount it to a vertical surface in every single use case. I apply a M/CB upon the entirety of the component to constrain it to primary planes, so that its Front Plane is the mounting surface, its Right Plane is the vertical centerline, and the Top Plane is its horizontal centerline. This is because we locate them vertically by their centerline height, and horizontally between each other as an on center dimension.
Every time that I wish to create a symmetric constraint (because there is none), I have to interrupt the M/CB command to take a measurement, divide by two, and re-edit the M/CB to apply the constraint as a distance. In my most common example above, I have to do this twice.
I'd very much like to see a Symmetric constraint added to M/CB feature's available constraints. Other "Advanced Mates" may or may not be as common or as applicable to constraints, as I have not considered the extensibility of this request.