Preheat oven to 350° F. Brush a 9" cake pan with butter. Line the bottom and sides with parchment paper.
Start by preparing the fruits. Zest the rind from the oranges (about 1 tablespoon), set aside. Then, prepare the fruits for the bottom of the pan. Slice the grapefruit, oranges, and clementine to ¼ to ½" thick rounds. Use a sharp knife to remove the rind from each citrus slice.
Heat the ⅓ cup sugar with 3 tablespoons water together. Stir to dissolve sugar. Pour this sugar mixture into the bottom of the prepared cake pan, arrange fruits slices on top. Set aside.
Next, let's prepare the cake batter. In a large mixing bowl, beat together the sugar, butter and orange zest until light and fluffy. Add in the eggs, yogurt, vanilla and orange juice and combine until well mixed.
In a medium bowl, sift and whisk together flour, baking powder and salt. Add the dry ingredients to the liquids ½ cup at a time. Mix until just combined.
When ready, pour batter over the citrus slices in the prepared cake pan. Spread the batter evenly. Gently tap the pan to remove any air bubbles. Pop into the preheated oven and bake for 40 minutes, or until the top turns golden brown. To check, insert a bamboo skewer or toothpick into the middle of the cake. It should come out clean. If not, bake for another 5 minutes.
Remove from oven, and place on cooling rack. Wait for cake to cool before inverting onto a cake platter or stand. And you are done! Stand back and admire your cool retro upside down cake.