Preheat oven to 350 degrees. Brush a 9-inch square baking pan with oil; line with a strip of parchment paper, leaving an overhang on two sides. Brush paper with oil. Sprinkle bottom of pan with 1/4 cup sugar. Trim pineapple wedges to no more than 4 to 5 inches long; arrange in pan in groups of four, alternating direction.
In a medium bowl, whisk together flours, baking powder, baking soda, and salt; set aside.
In a large bowl, whisk together buttermilk, eggs, vanilla, oil, and remaining 3/4 cup sugar. Add flour mixture; mix just until combined. Pour batter over pineapple. Tap pan firmly on counter to settle batter, then smooth top.
Bake until a toothpick inserted in center of cake comes out clean, 50 to 60 minutes. Cool 20 minutes in pan; invert onto a serving platter (peel off and discard paper).
Preparing Pineapple: Slice off top and bottom ends so pineapple sits level on board. Slice off skin in wide strips. Then, for this cake, cut pineapple lengthwise into thin wedges, and slice off core from the tip of each wedge. Cut into "sticks" and arrange in groups of four in baking pan, alternating direction.