1. Preheat oven to 400°F with rack in middle. Coat a 9-inch round cake pan with cooking spray.
2. Whisk together flour, baking powder, baking soda, and salt in a small bowl. Set aside.
3. Beat butter and 2/3 cup sugar with an electric mixer at medium-high speed until batter is smooth, about 2 minutes. Beat in the vanilla, egg, and lemon zest. Mix until well combined.
4. At low speed, mix in the flour mixture in 3 batches, alternating with buttermilk. Start with the flour and end with the flour. Mix until just combined. Don't over mix.
5. Spoon batter into cake pan, smoothing the top with a spatula. Scatter blackberries and rhubarb evenly over the top of the batter. Sprinkle with remaining 2 tablespoons of turbinado sugar.
6. Bake until cake is golden brown and a wooden pick inserted into center comes out clean, 20-25 minutes. Let the cake cool in pan for about 10 minutes, then turn out onto a rack and cool completely. Serve the cake upside down to show off the pretty berries and rhubarb. Cut into slices.