Pit and peel 2 avocados. Cut peel and white membrane from 2 oranges. Cut peeled avocados and oranges into 3/4-inch cubes and put in a bowl. Add rice, almonds, dates, and 1/3 cup dressing; mix.
Arrange spinach around rims of salad plates. Mound rice mixture in center of spinach.
Pit, peel, and thinly slice remaining avocado lengthwise. Cut peel and membrane off remaining oranges, cut fruit in halves, then slice crosswise. Arrange avocado and orange slices around rice mixture. Moisten salads with remaining dressing. Season to taste with salt.