Trim the rhubarb stalks to snugly fit into a pan and arrange on top of brown sugar to cover the base.
Whisk together the flour, baking powder, cardamom and salt in a medium bowl. Stir together the yogurt and remaining orange juice and zest in a small bowl.
In the bowl of a stand mixer or in a large bowl using a hand mixer, beat the butter and both sugars until pale and fluffy. Beat in the eggs, one at a time, until fully incorporated. Beat in the vanilla extract.
Add the flour mixture to the butter mixture in 3 additions, alternating with the yogurt and beginning and ending with the flour.
Transfer the batter to the pan, on top of rhubarb stalks, and smooth the top. Bake in oven for 1 hr. and 15min. The top should be browned and the edges bubbly. A cake tester inserted in the center will be clean.
Cool the cake for 5 minutes then run a small sharp knife around edges. Place a plate on top of the cake pan and invert. Slowly lift the cake pan away.