But the components are not mirrored?
When I say "Show only the top layer", it does mean only the top layer and nothing else, e.g. "display none top;". That way, components, pads, and other items are not copied as part of the group, and don't get mirrored. The group only contains the copper traces.
After renaming and when placed exactly under the top layer, any vias and pads will connect the top traces to the bottom copy automatically. Any missing connectivity will show up as airwires to be routed manually as usual.
Again, I'm not sure this is what the OP wanted to achieve, but I've used this method for increased current capacity between thru-hole components.
Another method, which works most of the time, is to paste the traces to be copied onto an empty board in an area that wouldn't overlap with anything in the real board. Then use "export-board.ulp" to create a script of the copied traces in "temp.scr". Then go back to the real board and execute "temp.scr". This should regenerate all the traces with the same names. Then group them, change the layer, and reposition the group as before.
This saves having to rename everything, but it doesn't always re-draw the traces correctly for me (like miters are sometimes forgotten). It will also display errors about non-existing layers and trying to create new signal names, but it will work anyway. "export-board.ulp" is in the user contributed ULP area on the web site. It could probably be fixed up to do this duplication task better. YMMV.