Preheat the oven to 350 degrees F.
Place the butter in an 8-by-8-by-2-inch square baking pan and set it over low heat to melt. When melted, sprinkle the brown sugar over the butter. Arrange the pineapple rings in a single layer on top of the sugar, making 3 rows. Cut the maraschino cherries in half, and place one half, cut-side up, in the center of each pineapple ring. Set the pan aside.
Sift the flour, baking powder and salt, then sift once more. Set aside. Using an electric mixer, mix together the shortening, granulated sugar, egg and vanilla. Blend in the flour mixture alternately with the milk, beginning and ending with flour. Stir only enough after each addition to combine.
Pour the batter carefully into the pineapple-lined baking pan and bake for 40 minutes. Test for doneness by inserting a toothpick in the center or pressing the cake lightly with a fingertip; if the impression springs back, the cake is done. Run a knife around the edges of the pan and place a serving dish on top. Invert the cake onto the serving dish. Leave the pan inverted over the cake for several moments to allow the syrup to soak into the cake.