Say all edges of a planar graph have been Tait colored 1,2,3 , then you can make Tait loops by following an edge colored 1 then 2 then 1 etc until the path closes into a loop.
Observe you may get another valid coloring by exchanging the 1 and 2 on this loop. Call this process TX. Of course TX affects the 13 and 23 loops, but I can show that the parity of the total Tait loop count remains unchanged by TX.
This then divides all colored graphs into odd and even modulo TX. I think this is a nice conserved parameter and shows previously unsuspected structure to the planar graphs.
Is it a new result?
