Combine all ingredients in a food processor. Process; scraping down sides to work in ingredients until all is smooth. Cover and refrigerate until ready to use. Clean out processor bowl.
In the food processor bowl, combine the onions, cilantro, oil, juice, zest, sugar, jalapeño, salt and pepper. Process, then transfer to a pyrex dish. Add the swordfish, cover, and marinate in the refrigerator, turning once or twice, for 1 hour.
Meanwhile, preheat the grill to medium-high. When the grill is ready, remove the fish pieces from the marinade (reserve the marinade), and lay them on the rack. Cover and grill, basting with the marinade and turning once, until the fish is done to your liking, about 8 minutes total for fish slightly undercooked at the center.
Transfer the fish to plates, top with a dollop of the mayonnaise, and serve immediately.
If you are preparing this recipe as gluten-free, just be sure to use a brand of mayonnaise that is known to be GF.
Nutritional information is calculated with the assumption that ALL of the avocado mayonnaise will be eaten. If you use a smaller amount, the calorie count will be lower.