Prepare the chicken--take out the bag of organs and neckbone that is tucked inside, then wash the cavity well with warm water.
To make the bird more attractive, cut off the triangular tail piece and the wing tips; remove any large clumps of fat under the skin that sometimes appear near the tail; pat the chicken dry with a paper towel.
Preheat oven to 425°.
Halve one of the oranges and squeeze out, and reserve the juice; quarter the rinds.
Do the same for the lime.
Put the orange quarters, lime quarters, and rosemary in the chicken's cavity; add as many orange and lime quarters as the cavity will hold.
Thinly slice the remaining orange into rounds.
Loosen the skin from the chicken; slip the rounds under the skin; leftover orange rounds can be used to garnish the dinner plates.
Whisk together the Worcestershire sauce, soy sauce, and honey; pour in the reserved orange and lime juices.
Place the chicken on a rack in a roasting pan; put it in the oven and cook for 20 minutes.
Decrease oven temperature to 350°; brush the chicken with the sauce; baste every 10 minutes until the chicken is done, about 40 minutes.
Remove the bird from the oven and let it rest for 15 minutes before carving.
Remove the skin and orange peels before serving.
Put the sauce in a small pot; bring to a boil and cook for 3 minutes; serve with the chicken.