Cook the rice: Put the rice into a medium saucepan. Run cold water into the saucepan, and with your hand, swish the rice around. Drain water into the sink. Repeat 3 more times, until water is more clear. Fill saucepan with 2 1/4 cups of water. Cover with lid. Cook rice over medium heat for 10 minutes. Lower heat to low and cook for an additional 10 minutes. Remove from heat, do not open lid. Let rice sit for 5 minutes or more.
While the rice is cooking, prepare the Spicy Mayonnaise. In a small bowl, whisk together the ingredients and set aside.
Cut the cucumber, tuna and avocado into 1/2" dice. In a small bowl, toss the avocado with the lemon juice, and cover with plastic wrap until ready to eat.
When the rice has cooked and rested, open lid and transfer the rice to a large bowl. Using a spatula, gently fold and lift the rice to allow steam to escape and to cool. Do not mash the rice or mix too vigorously (try not to break the rice kernels). After much of the steam has been released, dribble in a little of the seasoned rice vinegar, fold and lift rice to distribute. Repeat with rest of rice vinegar. Keep rice covered with a damp cloth until ready to serve. Do not refrigerate.
To serve, divide the rice between 4 bowls. Top with cucumber, avocado, then tuna. Spoon a dollop of the Spicy Mayonnaise. Top with shredded seaweed. Serve with soy sauce and wasabi at the table.