Place beets in a medium saucepan. Add enough water to cover and a generous amount of salt. Bring to a boil over high heat and cook until beets are tender and easily pieced with the tip of a paring knife, about 1 hour. Drain and rinse beets under cold water; peel and pat dry. Cut beets in half lengthwise and thinly slice on a mandoline.
Place a 9-by-6-inch piece of parchment paper on your work surface, with short ends parallel to edge of work surface. Starting at the top of the parchment paper, lay slices of yellow beets across the top of the parchment, slightly overlapping one another, and extending over the top and sides of the parchment. Using the red beets, place a second row of beets, slightly overlapping each other, so that they overlap the layer of yellow beets by half, and extend over the sides of the parchment. Continue layering process, alternating yellow and purple beets, until parchment is covered with beets. You may not need to use all the beets. Trim beets to size of parchment paper.
Cover beets with an 11-by-8-inch sheet of parchment paper and carefully invert beets onto second piece of parchment. Remove first piece of parchment and discard. Using a very sharp knife, cut layered beets lengthwise on parchment into 4 equal strips; set aside.
Place orange peel and juice in a small saucepan over medium heat. Cook until juice is reduced to about 2 tablespoons. Transfer orange mixture to a medium bowl and whisk in olive oil, lemon juice, and mustard; season with salt and pepper. Set vinaigrette aside.
Place celeriac and apples in a medium bowl; drizzle over 3 tablespoons of vinaigrette and season with salt. Toss until well combined.
Carefully invert one of the strips of beets onto each of 4 serving plates; peel off parchment and discard. Divide celeriac mixture between the 4 plates. Top each with 3 1/2 ounces crab meat. Garnish with mache and orange peel; drizzle with remaining vinaigrette. Serve immediately.