Congratulations to Kristina Frizyuk!
It turns out that an elegant condition (eq. 17) describes all possible symmetries of the nanoparticles and their crystalline lattices, harmonic orders, and angular momenta of vortex beams. In turn, another simple condition (eq. 105) outlines all possible cases when excited by vector beams. Despite the simplicity of the answer, various nuances arise in the process of deriving it, which are detailed in the paper.