Cook the rice: In a small pot, combine the rice and water and place over high heat. When it boils, cover tightly and reduce the heat to low. Cook for about 40 minutes, until the water is absorbed. Take off the heat and let stand, covered, for 10 minutes. Stir together sugar and 1 tablespoon of vinegar, drizzle over rice and fold in.
In a small bowl, combine the radish and remaining rice vinegar and toss to mix. Let stand to marinate for 15 minutes.
In a small bowl, combine the mayo and wasabi and stir. If you really like wasabi, add more to taste. Trim the avocado to the length of the nori, then steam just until crisp-tender. Cool.
Slice the avocado in half, then remove the pit. Slice the avocado fruit in the shell lengthwise to make 8 slices. Scoop out carefully with a spoon.
Lay out four tortillas, then spread 1/4 of the rice on one side of each tortilla. Smear 1/4 of the mayo mixture across the rice, then place the asparagus, avocado slices, pickled ginger, and kraut in rows alongside. Roll up like sushi, starting at the side closest to you. Slice in half and use toothpicks to hold the roll closed. Serve or wrap and refrigerate.