Preheat oven to 350 degrees F (175 degrees C). Grease two 9-inch round cake pans.
Finely grate zest from 2 clementines and set aside. Juice all 4 clementines into a measuring cup; add enough water to reach 1 cup.
Pour clementine juice into a large bowl. Add zest, cake mix, eggs, oil, orange extract, vanilla extract, salt, and red food coloring. Blend with an electric mixer until batter is smooth, about 2 minutes. Divide batter into greased cake pans and smooth the top.
Bake cakes in the preheated oven until a toothpick inserted into the centers comes out clean, 24 to 29 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack, about 30 minutes more.
Transfer cakes to a flat surface. Carefully trim the tops of the cakes with a sharp knife to level them; cut each layer in half horizontally. Transfer one cake layer to a serving plate and spread with 1/2 cup of the frosting; top with another cake layer and 1/2 cup of frosting. Scatter chopped clementine over the frosting. Stack and frost third and fourth cake layers. Arrange clementine wedges around the perimeter of the top layer.