For the cake: Preheat the oven to 350°F
Sift the flour twice with the cocoa powder and 1/2 cup of the sugar.
With an electric mixer on high speed, beat the egg whites, cream of tartar and salt until soft peaks form when the mixer is removed from the batter.
Add half of the remaining sugar and beat for 1 minute. Add the remaining sugar, 2 tablespoons at a time, beating after each addition.
Stir in the vanilla.
Fold the flour and sugar mixture into the egg whites, 1/4 cup at a time, until just incorporated.
Put the batter in an ungreased 10" tube pan and bake until the cake is springy to the touch, about 1 hour. Invert the pan and let the cake cool completely before removing from the pan. (This can be made in advance and stored in an airtight container for up to 5 days.)
For the icing and oranges: Whisk the sugar, milk and vanilla or rum extract together in a mixing bowl. Stir in the coconut.
Drizzle the icing over the cake.
Serve each slice of cake with orange sections arranged like a fan on the side of each piece.