Transverse patterns in polariton fluids were recently studied as promising candidates for all-optical low-intensity switching. Here, we demonstrate these patterns in a specifically designed double-cavity system. We theoretically and experimentally analyse their formation and optical control. Our detailed theoretical analysis of the coupled nonlinear dynamics of the optical fields inside the double-cavity and the excitonic excitations inside the embedded semiconductor quantum wells is firmly based on a microscopic many-particle theory. Our calculations in the time domain enable us to study both the ultrafast transient dynamics of the patterns and their steady-state behavior under stationary excitation conditions. The patterns we report and analyze go beyond what can be observed and understood in a simple scalar quantum field. We find that polarization-selective excitation of the polaritons leads to a complex interplay between longitudinal-transverse splitting of the cavity modes and the spin-dependent interactions of the polaritons' excitonic component.