Sorry Bradtherad, the flux in a transformer does not know in which winding the current is flowing...
Right, since it converts 12W in one direction, then it should be able to convert 12W in the other direction. That is, theoretically, and if nothing else interferes.
Here is how I see it:
In the first diagram...
The transformer converts our specified 12W. Its primary side shows an impedance of 4400 ohms to the electric company (at 230 VAC).
In the second diagram...
The transformer delivers 230 VAC through 2 x 4400 ohms total impedance. The load does not get the entire 230 VAC. Rather it gets half of that.
I think the transformer will need a much higher power rating, in order for the primary to have low enough impedance for the load to get sufficient voltage across it.
Of course it is essential that we factor in DC resistance of the winding. I just measured several small wall wart adapters, using my ohmmeter. The primaries range from 100 to 500 ohms. This by itself will reduce voltage on a 4400-ohm load hooked up in series.
Transformer theory is complicated, and I realize I could be wrong. I once tried to make a homebrew inverter, starting with a big step-down transformer, 4 inches on each side. I could barely get a 7W bulb to light. I believe it was because the transformer primary had high internal impedance.