From an end user's point of view, there is no difference in the orientation. But from the device's point of view, one orientation has pins 1-8 and the other has pins 8-1. It's up to the device to figure out which is being used. This is still considered symmetrical though.
I get the point you're trying to make, but the connector is electrically symmetrical: "made up of exactly similar parts facing each other or around an axis"
And then some - hadn't really thought about it until your comment, but somehow we regressed from infinite orientations (well, in that plane) to one, and are now all excited about having two...
Are there any example of barrel-style connectors with >2 pins, i.e. coaxial barrels? Similar is the 'jack' (3.5mm and such) but with split regions along a single pin rather than concentric rings of course. I wonder why such designs aren't more popular/explored (that I'm aware of anyway) for multi-pin data connections?
But usually the inner one was only connected via a resistor, to tell the device the wattage of the charger (but could be used by a bunch of 1 wire protocols instead.