Can a 6 volt positive ground system share a common chassis ground with a 12 volt negative grd
As you found out yourself, you need a inverting voltage converter for that.

-6V → +12V. They share the same ground level.







simulate this circuit &ndash; Schematic created using CircuitLab

And the answer to your questions is: yes, sure.

EDIT, as asked for in the comments:

There is nothing special about ground other than we define it to have a potential of 0V. It's just a reference potential. And voltage isn't more than the difference between the potential of two points in the circuit. 

So, this -6V is a property of the in wire and this +12V is a property of the out wire. Ground does not see anything what's going on on other wires.