I am currently listening to a podcast about beginning of WWI and how the French attacked through Alsace-Lorraine to (initially) little resistance while the Germans were sweeping through Belgium. It made me think that if the Germans went straight for Paris and the French for Berlin they could have taken each other's capitals. Was there ever a battle in history where two armies outflanked each other in a similar way and just ended up swapping territories?
Not quite on a strategic level, but this could happen in Classical Greece if the two armies happened to choose the same tactical plan.
One of the basic Greek approaches to battle was to line up all the heavy infantry with the strongest troops and the commander on one wing. They often chose to put the commander on the right, because the natural drift of the phalanx to the right gave that wing a natural advantage. As the phalanxes advanced, both would slowly drift to the right, which meant that the right wing ended up overlapping the enemy left. The result would be an outflanking and local victory. However, if the enemy had followed the same plan, both sides would win a partial victory, with both sides' left wings being quickly routed.
This is exactly what happened at the battle of Laodokeion, fought between the Tegeans and the Mantineians in 424/3 BC. Both right wings routed their opponents; both sides claimed overall victory, set up a trophy, and dedicated spoils to Apollo at Delphi.
Something similar happened at the battle of Kounaxa, in 401 BC, between King Artaxerxes II and the pretender Kyros the Younger. While the left wing of Kyros' army was quickly outflanked and routed, and he himself was killed in the centre, the Greek mercenaries on his right wing encountered no resistance, and thought they were winning the battle. They eventually realised that their camp was being plundered and that there was a large force of Persian cavalry in their rear. They were then forced to turn 180 degrees to face the threat.
At the battle of Koroneia, in 394 BC, between Spartan and Theban coalition armies, things went a step further than this. Again, both sides put their best troops on the right, and won the battle easily there. The Spartans were already celebrating victory when someone told their commander, king Agesilaos, that the Thebans were plundering the Spartan baggage train. When the Thebans discovered that all their allies had fled, they turned around and tried to make their way back to the hillside where friendly forces had sought refuge - but they found the Spartans in their way. With the sides reversed, the Thebans and Spartans fought a second battle, which, as Xenophon tells us, was bloody and pointless. Agesilaos nearly died in the fighting. Still, the Thebans were roughly handled in the melee, allowing the Spartans to claim victory after all.
In 1973, the Israelis bypassed the bulk of the Egyptian Army and rolled up the West Bank of the Suez Canal. This resulted in the two Armies facing each other down, each controlling the "wrong" bank. The Egyptians were in the Israeli forts of the Bar-Lev line, and the Israelis were in the Egyptian jumping-off points for the invasion.
Almost. The Israelis still had a supply line and crossing point back to Israel down the South, the Egyptian forces were actually reliant, after the ceasefire, on the Israelis letting supplies through their lines, while the Israelis could supply themselves.