Trim off the ends of the sugarcane and cut them into skewers. Rinse the shrimp under cold running water and then drain and blot dry with paper towels. Skewer the shrimp on the sugarcane.
Place the brown sugar, butter, rum, mustard, vinegar, cinnamon, cloves, pepper, and salt in a non-reactive saucepan and bring to a boil over high heat. Reduce the heat to a simmer and continue to cook until the mixture is thick and syrupy, about 3 to 5 minutes. Taste for seasoning, adding salt and pepper as necessary. Set aside a small amount of the glaze to be used for dipping the cooked shrimp. The recipe can be prepared up to 1 hour ahead to this stage; keep the shrimp in the refrigerator, covered. The glaze can be kept at room temperature.
Preheat the grill to high. When ready to cook, brush and oil the grill grate. Place the shrimp kebabs on the hot grate and grill until cooked, 2 to 4 minutes per side. Generously baste each with glaze once before turning and once before removing from the grill. The shrimp will be firm and white when cooked through. Serve the reserved glaze as a sauce on the side.