When nanomaterials are put in contact with living organisms, their interaction with biomacromolacules and biological barriers will determine their bioactivity, biological fate and cytotoxicity. In this context, the role of symmetry/shape anisotropy of both the nanomaterials and the biological interfaces on their interaction mechanism, is a relatively unaddressed issue. Here, we study the interaction of gold nanoparticles (NPs) of different shape (nanospheres and nanorods) with biomimetic membranes of different symmetry, i.e. flat membranes (of 2D symmetry, representative of the most common plasma membrane geometry), and cubic membranes (of 3D symmetry, representative of non-lamellar membranes, encountered in Nature in peculiar biological conditions). Through an ensemble of structural techniques (Neutron Reflectometry, Grazing Incidence Small-Angle Neutron Scattering), we found that, on a nanometric lengthscale, the structural stability of the membrane towards NPs is dependent on the topological characteristic of the lipid assembly and of the NPs, with higher symmetry related to higher stability. In addition, Confocal Laser Scanning Microscopy analyses highlight, on a micrometric legthscale, that cubic and lamellar phases interact with NPs according to two distinct mechanisms, related to the different structures of the lipid assemblies. This study represents a first attempt to systematically study the role of membrane symmetry on the interaction with NPs; the results will contribute to improve the fundamental knowledge on nano-bio interfaces and, more in general, will provide new insights on the biological function of non-lamellar cubic arrangements of interfacial membranes as a response strategy.