It’s easier to see that this circuit is rather symmetric when you redraw it

Image

Then with both switches closed blue wire (the one in the middle) should only carry small current. If both halves are exactly the same it should be zero

In general when circuit doesn’t include nonlinear components (semiconductors most commonly) you can sum two valid solutions and you’ll get another valid solution. Note that when one switch is closed, current in blue wire goes one way and when the other is closed, in opposite way