How to get +6V and -6V out of 12V PSU
Ground is an illusion.  We choose a node and decide we're going to call it 0V, and give it the magical name ground.  If you create a reference of 6V from your 12V supply, and call it ground, your 12V supply just became +/-6V.

Of course, ground is also assumed to be low impedance, so you'll have trouble if you create it using a 100K voltage divider and then try to use it as a current return.  Because of that, we generally call it Vref, and thus remember to take into account its current limitations relative to the supply.

If there's a DC offset between Vref and your I/O, capacitors will resolve it nicely.