In a three-dimensional universe, wires can cross over each other without connecting. But on the two-dimensional playfield, a cross requires a planar device that mutually exchanges inputs. That device implements the XOR swap algorithm. It stores the XOR of inputs A and B into a temporary variable, C:

C = A ⊕ B

Then it XORs the inputs with that temporary variable, which swaps them:

A ⊕ C = A ⊕ A ⊕ B = B

B ⊕ C = B ⊕ A ⊕ B = A

Those steps suggests this circuit:

Since two-dimensional XOR gates exist and the circuit does not contain any cross-overs, it achieves a planar swap.

In following actualization, the lower XOR gate of the schematic is made from three NORs and a horizontal I-tetromino operating as an OR. The other pieces function as wires.