When placed over a less dense medium, a liquid layer will typically collapse downwards if it exceeds a certain size, as gravity acting on the lower liquid interface triggers a destabilizing effect called a Rayleigh-Taylor instability1,2. Of the many methods that have been developed to prevent the liquid from falling3-6, vertical shaking has proved to be efficient and has therefore been studied in detail7-13. Stabilization is the result of the dynamical averaging effect of the oscillating effective gravity. Vibrations of liquids also induce other paradoxical phenomena such as the sinking of air bubbles14-19 or the stabilization of heavy objects in columns of fluid at unexpected heights20. Here we take advantage of the excitation resonance of the supporting air layer to perform experiments with large levitating liquid layers of up to half a litre in volume and up to 20 centimetres in width. Moreover, we predict theoretically and show experimentally that vertical shaking also creates stable buoyancy positions on the lower interface of the liquid, which behave as though the gravitational force were inverted. Bodies can thus float upside down on the lower interface of levitating liquid layers. We use our model to predict the minimum excitation needed to withstand falling of such an inverted floater, which depends on its mass. Experimental observations confirm the possibility of selective falling of heavy bodies. Our findings invite us to rethink all interfacial phenomena in this exotic and counter-intuitive stable configuration.