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@hamrack
I think the purpose of the crossover is to increase the torque applied to each arm of the brake (and hence braking force) by increasing the perpendicular distance from the pivot (ie the distance to the pivot measured perpendicular to the direction of the applied force). It'll function without it, but not quite as well. I seem to remember dia-compe do a U-brake?
Edit: I can't find it for sale but CRC have one.