Prepare the upside down topping: Prepare a 9 inch springform pan by lining it with parchment paper. Grease the parchment paper with butter, and sprinkle wth sugar. With the skin still on, slice the mandarins in 1/4 inch slices. Use a paring knife to carefully remove the skin from the citrus slices (like cutting off the outer ring). Doing it this way will ensure your mandarin slices stay intact and does not break apart when slicing them. Line the prepared pan with the orange slices.

Set aside. Make the cake batter: Preheat the oven to 350F. In a food processor, process the whole mandarins (with skin) until almost pureed. Set aside. Place the sugar and eggs in a large bowl and beat with a mixer until light and fluffy. Add the baking powder to the bowl, along with half the flour. Continue mixing, then add in the rest of the flour. Add in the softened butter and mix until completely blended. Add the yogurt, pureed mandarins, and vanilla extract to the bowl. Mix one more time. Pour the batter over the mandarins. Bake the cake for 40-45 minutes, or until a toothpick inserted comes out clean. Allow the cake to cool completely. When ready to serve, flip upside down to turn out the cake onto a large plate. Peel the parchment paper away from the mandarins and serve