Heat gas grill to medium-high or coals in a charcoal grill to medium-hot. Lightly coat grill rack with oil or cooking spray.
In a small bowl, stir together mayonnaise and 1 tablespoon of the hoisin sauce. Cover and refrigerate until ready to serve.
In a large bowl, mix together pork, mushrooms, scallions, soy sauce, remaining tablespoon hoisin sauce and ginger. Form into 6 equal-size patties. Grill for 3 minutes per side or until internal temperature registers 160 degrees F on an instant-read thermometer. Remove to a plate and keep warm.
To serve, spread 1 tablespoon of the mayonnaise mixture on one side of a tortilla. Place arugula and burger on top and fan radish slices over burger. Fold over.