Nonlinear phenomena are central to modern photonics but, being inherently weak, typically require gradual accumulation over several millimeters. For example, second harmonic generation (SHG) is typically achieved in thick transparent nonlinear crystals by phase-matching energy exchange between light at initial, ω, and final, 2ω, frequencies. Recently, metamaterials imbued with artificial nonlinearity from their constituent nanoantennas have generated excitement by opening the possibility of wavelength-scale nonlinear optics. However, the selection rules of SHG typically prevent dipole emission from simple nanoantennas, which has led to much discussion concerning the best geometries, for example, those breaking centro-symmetry or incorporating resonances at multiple harmonics. In this work, we explore the use of both nanoantenna symmetry and multiple harmonics to control the strength, polarization and radiation pattern of SHG from a variety of antenna configurations incorporating simple resonant elements tuned to light at both ω and 2ω. We use a microscopic description of the scattering strength and phases of these constituent particles, determined by their relative positions, to accurately predict the SHG radiation observed in our experiments. We find that the 2ω particles radiate dipolar SHG by near-field coupling to the ω particle, which radiates SHG as a quadrupole. Consequently, strong linearly polarized dipolar SHG is only possible for noncentro-symmetric antennas that also minimize interference between their dipolar and quadrupolar responses. Metamaterials with such intra-antenna phase and polarization control could enable compact nonlinear photonic nanotechnologies.