Is there a way to "mirror" an empty? Using Blender 2.8, and I'm using a number of empties to drive some animations within in a collection of objects in a parented structure.

I can mirror objects using modifiers, and I know modifiers are not available for empties, but is there a way to "mirror" axis empties in some similar automated way? 

My workflow so far has been manual in a way to first dupe the empty (has the benefit of also copying the animation for it too), then manually changing the xyz loc/rot values -1 as needed to get them to be "mirrored". This was OK when there were a few empties, but as you can guess as the project has grown in scale and complexity this is becoming very time consuming, and the manual process in risky and prone to lots of errors.

Is there a trick/method to "mirror" empties for a more automated process to reduce time and risk/errors?
Constraints may suit. The copy Xform constraints have the option to invert.

Here we have 2 axis-Empties, both parented to the cube-Empty in the middle. The right Empty is constrained to the left Empty as follows:



And it's parented to the cube Empty Without Inverse.

The arrows are parented to their respective axis-Empties, for illustration:



If the Empties are all ShiftD duplicated together, Blender is smart enough to keep the relations between the duplicates, and the whole rig can be moved/rotated using the cube.