Special equipment: a well-seasoned 10-inch cast-iron skillet
Preheat oven to 350 degrees F.
Cut pineapple crosswise into 3/8-inch-thick pieces. Melt butter in skillet. Add brown sugar and simmer over moderate heat, stirring, 4 minutes. Remove from heat. Arrange pineapple on top of sugar mixture in concentric circles, overlapping pieces slightly.
Sift together flour, cardamom, baking powder, and salt. Beat butter in a large bowl with an electric mixer until light and fluffy, then gradually beat in granulated sugar. Add eggs, 1 at a time, beating well after each addition. Beat in vanilla and rum. Add 1/2 of the flour mixture and beat on low speed just until blended. Beat in pineapple juice, then add remaining flour mixture, beating just until blended. (Batter may appear slightly curdled.)
Spoon batter over pineapple topping and spread evenly. Bake cake in middle of oven until golden and a tester comes out clean, about 45 minutes. Let cake stand in skillet 5 minutes. Invert a plate over skillet and invert cake onto plate (keeping plate and skillet firmly pressed together). Replace any pineapple stuck to bottom of skillet. Sprinkle rum over cake and cool on plate on a rack.
Serve cake just warm or at room temperature.
Some of the food editors found 3 teaspoons of cardamom to be too much, but others loved the intense flavor. Cake may be made 1 day ahead and chilled, covered. Bring to room temperature before serving.