Homing position has nothing to do with the direction of the axis. You need to set up your home switches to accommodate the machine moving in the correct directions.
As for the DRO's, they'll read exactly what the g-code tells them too. If you want to run parts in negative coordinates, they need to be programmed that way. The Roadrunner is programmed in positive coordinates, so that tells me that your machine was running backwards.
If you have your home position in the back right, you can't have X+ go left, and Y+ move towards you.
You can use home offsets to set your home switches to be the max positive coordinates, which would make your 0,0 at the lower left.
As Jeff says, a pic or drawing of the machine layout would be a big help.