To make the cake: Preheat oven to 375 degrees. Pour egg whites into a large bowl and add cream of tartar, vanilla, and salt. Beat with an electric mixer at high speed until soft peaks form. (Egg whites are sufficiently beaten if they stay put when you shake the bowl.) Sprinkle in sugar, 2 tablespoons at a time, continuing to beat at high speed until sugar dissolves and egg whites stand in stiff, glossy peaks. (Do not scrape the bowl.) With the mixer on low speed, gradually beat in flour until just blended. Wash and dry the beaters.
In another large bowl, beat egg yolks at high speed until they are thick and lemon colored. With a wire whisk, fold in half of the egg white-flour mixture and grated orange and lemon zest.
Drop a heaping tablespoon of white batter into an ungreased 8-inch tube pan. Then drop a heaping tablespoon of yellow batter next to white batter. Repeat this process, forming a checkerboard pattern, until pan is three-quarters full. (Bake remaining batter in 4 individual 6-ounce custard cups for 15 minutes.) Bake cake 30 to 35 minutes or until a toothpick poked in center of cake ring comes out clean. Invert cake over a cooling rack (do not remove from pan); cool 1 hour.
Meanwhile, prepare frosting. Beat butter and lemon and orange zests at medium speed with an electric mixer until creamy. Gradually add the powdered sugar, alternating with lemon and orange juices, beginning and ending with powdered sugar. Beat until spreading consistency.
To remove cake from pan: Gently run knife between cake and pan, both inner and outer rings. If you have difficulty, use your hands to gently pull cake out; it should spring back if you press it in. Frost cake and refrigerate. Before serving, garnish with fresh tulip petals. (Tulip petals are edible, though here we have used them strictly decoratively.)