I have an object with its unique axis, and I have mirrored the object across XY coordinates, then have made the object editable. I have split the result into individual objects. The original has the correct axis, however the mirrored "split" object retains the originals axis coordinates. Of course I could "center axis to" but this would not be correct. I could also tweak and get it "pretty close" using the axis tool, but I don't want "pretty close" I want the exact mirrored axis.
One way to do this is to copy your object and then, in the coordinates manager scale section, type a minus in front of the 1 on the axis on which you want to mirror, and click apply. Then type a minus sign in front of the x/y/z position parameter and click apply.
That should take care of the mirroring. Note, however, that it also reverses the polygon normals, so you should select them and reverse them back.