Place the cooked sushi rice in a small bowl and gently fold in the rice vinegar until well combined.
Slice the sushi-grade salmon and cooked shrimp into long, thin strips.
Julienne the cucumber and slice the avocado into thin wedges.
Place one sheet of nori on a bamboo rolling mat, shiny side down.
Wet your fingers slightly and spread half of the rice over the nori, leaving a 1-inch border at the top.
Arrange half of the salmon, shrimp, avocado, and cucumber in a horizontal line across the center of the rice.
Using the mat, roll the nori tightly over the fillings, applying gentle pressure to seal the roll at the border.
Repeat the process with the second sheet of nori and the remaining ingredients.
Use a very sharp, damp knife to slice each roll into 6-8 pieces.
Serve the rolls immediately with tamari, wasabi paste, and pickled ginger on the side.