Reconstruction of differential signal
If I understand correctly, you have  3 voltages, -12V , 0V, +12V. You then wish to translate it to 0V , xV, +12V and then those voltages back to  -12V , 0V, +12V. If so, you can bring your input to op amp that is connected as inverting amplifier which has an offset voltage on its non inverting input. That way you can get voltages between 0V and +12V. Then do the same step to get -12V , 0V, +12V. (Output from op amp to second op amp)
This is the circuit (UPDATED)


Red [Vi] - original signal
Blue [Vo1] - 0V - +12V
Green  [Vo2] - -12V - +12V  --&gt; Reconstructed