In a small glass bowl, combine the cooked sushi rice with the rice vinegar and coconut sugar, folding gently with a spatula to season the grains without crushing them.
Place the nori sheet on a bamboo rolling mat and spread the seasoned rice in an even, thin layer across the surface, leaving a one-inch margin at the top edge.
Slice the salmon, shrimp, cucumber, and avocado into long, thin strips.
Arrange the salmon, shrimp, cucumber, and avocado strips in a horizontal line across the center of the rice-covered nori.
Lightly dampen the top margin of the nori with water, then use the bamboo mat to roll the seaweed tightly over the fillings into a firm cylinder.
Using a very sharp, slightly wet knife, slice the roll into eight uniform pieces and serve immediately with tamari, wasabi, and pickled ginger on the side.