Preheat oven to 325 degrees. Coat a 9-inch by 13-inch cake pan with nonstick spray.
In a large bowl, whisk the eggs and oil together until well blended. Whisk in the cake mix until smooth. Carefully fold in the mandarin oranges and their juice.
Pour the batter into the prepared pan and bake 25 to 35 minutes (check at 25 minutes). Cool completely, at least 30 minutes.
When the cake is cool, blend whipped topping and dry vanilla pudding mix in a large bowl (do not prepare the pudding mix into pudding). Carefully fold in pineapple and juice. Spread the frosting in a thick, even layer on the cake.
Cover and refrigerate at least 30 minutes before serving. Refrigerate any leftovers.
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Notes
Oranges: To substitute fresh oranges, tangerines, or clementines (more work than canned oranges), cut the segments from their membranes into supremes, and squeeze the extra juice from the fruit for making the cake.
Whipped topping: To substitute homemade whipped cream for the whipped topping, whip 3 cups of heavy cream with 6 tablespoons powdered sugar and 1 1/2 teaspoons vanilla extract. For best results, chill metal mixing bowl and metal whisk in freezer for 10 to 15 minutes for before whipping. Whipped cream can be stored covered, in the refrigerator, for 10 hours.
Pineapple: To substitute fresh pineapple, pulse chunks of pineapple in a food processor. Drain well and reserve the juice for the frosting. Note: Do not use fresh pineapple with a gelatin-based pudding. Bromelain, the natural enzyme found in fresh pineapple, will break down gelatin, which can soften the frosting if you use a gelatin-based pudding.
Yield: This recipe makes one 9-inch by 13-inch cake. I usually cut 20 generous slices, but you can cut them bigger or smaller depending on your preference.
Storage: Store leftovers covered in the refrigerator for up to 4 days.