I was able to do this cutting the text and pasting it in a safe place, then cutting the path and pasting in Illustrator. Mirror the path in Illy, then copy and paste back into ID and paste your dsaved text back on the new path.
In place of pasting it in Illustrator and then MIRRORING it,
Once you have cutted the text and pasted it in a safe place , select the object and create a copy of it in the same document , select the copy and "flip horizontal" or "flip Vertical" from "Object-->Transforms"
choosing the "flip horizontal" or "flip Vertical" depened on your image and more on how you want to create the mirror image.
I have managed to do that by:
1) Selecting type and path (all nodes) with direct select tool (white arrow)
2) Clicking Flip Horizontally button in Control Panel (you can also click additionally Flip Vertically if you want, depending what kind of mirroring you want to do)
3) Choosing Object > Paths > Reverse Path
Yes, that does work.
I wonder why selecting with the direct select tool works, but the regular selection tool does not (I tried reversing the path as the first test after flipping the path normally).
Peter, I had a play and it seems like if you use the Selection tool it affects the object as a whole where the Direct Selection tool affects only the frame.Do a test with a wonky shape and as you flip with both tools, you will see that the Blue anchored box remains in place with the White pointer (ie does not flip).