Step 1Heat the oven to 350 degrees.
Step 2In a large 10- to 12-inch skillet, preferably cast iron, melt the butter over medium heat. Stir in the brown sugar and cook, stirring constantly, until the sugar begins to bubble. Remove the pan from heat. If you are baking the cake in the skillet, set the whole thing aside to cool. If you are baking the cake in a springform pan, scrape the melted sugar into a 9-inch round pan (line the base with parchment paper, to about one-half inch up the sides of the pan). Spread the sugar mixture to the edges and set pan aside.
Step 3Once the sugar and butter mixture has cooled, arrange the apricot pieces in a circular pinwheel design on top of it. Scatter any remaining pieces on top of the first in a small second layer. (The apricots will cook down considerably in the oven, so you want to be generous with the fruit.)
Step 4In a medium bowl, combine the flour, cornmeal, baking powder and salt. In a separate large bowl, and using an electric mixer, beat the sugar and eggs until the eggs have doubled in volume and the mixture is pale and golden, 3 to 5 minutes.
Step 5Working in small batches and alternating with the oil, sift the dry ingredients into the wet ingredients: Begin by adding just a bit of the dry ingredients and stir until just combined, then add a bit of the olive oil and stir until just combined, and so on, until all of the ingredients are incorporated. Stir in the orange juice, orange zest and orange flower water. Scrape and smooth the batter into the prepared pan so that it covers the pinwheel of apricots.
Step 6Bake the cake until it is golden and springy to the touch and a knife or toothpick inserted into the center comes out clean, about 35 minutes (a springform cake will require additional time). Allow the cake to cool for 5 minutes. Then, if using a skillet, place a large plate over the top and carefully flip the pan so that the cake inverts onto the plate with the apricots on top. There may be some excess caramel that dribbles over the edge -- take care not to burn yourself. (If using a springform, place a plate over the pan, flip over the pan, release the spring mold and peel back the parchment.) The cake will be best on the day it is made. Serve it slightly warm or at room temperature.