Not sure what you mean by a "brake axle"?

IF the wheel is dished properly, you can install it either way and it will remain centered.
If NOT, it'll be offset to one side or the other by the same amount.
If the fork is off, the wheel will stay to the same side. (if dished correctly)
It sounds like you have a little of both problems.

If your truing stand only has one "indicator", you have to flip the wheel in the stand and make sure you have the same distance from the indicator in either orientation. (You can't assume the indicator is correct until proven)