Slice a long cut into the side of each swordfish steak, forming a deep pouch. Reserve.
In a large shallow platter, whisk together wine, lime juice, soy sauce, mustard, ginger, pepper, sesame oil, and garlic. Arrange steaks in liquid, flipping to fully coat. Wrap with plastic wrap and chill for 30 minutes.
Take steaks out of marinade, saving liquid. Season insides of steaks with salt and reserve.
Toss arugula in reserved liquid, fully coating. Take arugula out of marinade. Spoon arugula into pouches of steaks, filling about 3/4 of the way. Fasten with toothpicks.
Lay steaks on a cooking spray coated grill rack and cook 5 minutes per side; fish should be tender and flake easily.