2. In a small (1 to 1 1/2-quart) saucepan, simmer the beets in salted water to cover until tender, about 20 minutes, depending on size. Cool to lukewarm, then cut into small sticks and place in a large bowl.
3. Mix together the chopped orange zest, lime juice, orange juice, salt, sugar and olive oil in a large bowl. Pour over the beets and let stand for 1 hour.
4. Peel away the brown skin and fibrous exterior layer of the jicama (a small knife works best for this), then cut in half. Lay each half on its cut side and slice 1/4-inch thick; cut slices into small sticks.
5. Cut stem and blossom ends off the remaining 2 oranges, stand on a cutting board and working close to the flesh, cut away the rind and all white pith. Cut between each white membrane to remove the segments.
6. Just before serving, add the jicama and most of the orange segments (save a few for garnish) to the beets; taste the mixture and season it with additional salt, if needed.
7. Lay out the lettuce on a serving platter. Scoop the salad into the center, then sprinkle with the peanuts and reserved orange segments. Garnish with the sugar cane and candies and the colorful salad is ready to serve.
For advance preparation: The beets will keep in the refrigerator for 3 to 4 days; let them come to room temperature before using. The oranges, jicama and sugar cane can be cut up and refrigerated the night before serving.
*A note about sugar cane: If you haven't worked with it before, don't be anxious. It's easy (though it takes a big sharp knife to peel), because it has a loose grain like balsa wood. You can simply shave off pieces to sliver. Your guests will enjoy picking up the pieces to chew on (you can't eat them); they have a delightfully fresh sweetness.