Preheat oven to 350°F.
Use a well-seasoned 10-inch cast-iron skillet. If you lack a cast-iron skillet of this size, make the caramel in a small pot and scrape it into the bottom of a similar-sized cake pan before arranging the pineapple slices. (I used a 9″ cake pan.)
Cut pineapple into 3/8-inch-thick slices.
Melt butter in skillet and add brown sugar. Simmer over moderate heat and slowly stir for about 4 minutes. Remove from heat.
Arrange pineapple slices on top of sugar mixture in concentric circles, overlapping pieces slightly.
Sift together flour, baking powder, and salt.
Beat butter in a large bowl with an electric mixer until light and fluffy. Beat in granulated sugar.
Add eggs, one at a time, beating well after each addition.
Beat in vanilla and rum.
Add half of 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 lumpy.)
Spoon batter over pineapple topping and spread evenly. Bake cake until golden and a tester comes out clean, about 45 minutes. Let cake stand in skillet for five minutes to cool slightly. Invert a plate over skillet and invert cake onto plate (keeping plate and skillet firmly pressed together).