We predict that the Lee-Huang-Yang effect makes it possible to create stable quantum droplets (QDs) in binary Bose-Einstein condensates with a heterosymmetric or heteromultipole structure, i.e., different vorticities or multipolarities in their components. The QDs feature flat-top shapes when either chemical potential, μ1,2, of the droplet approaches an edge of a triangular existence domain in the (μ1,μ2) plane. QDs with different vorticities of their components are stable against azimuthal perturbations, provided that the norm of one component is large. We also present multipole states in which the interaction with a strong fundamental component balances the repulsion between poles with opposite signs in the other component, leading to the formation of stable bound states. Extended stability domains are obtained for dipole QDs; tripole ones exist but are unstable, while quadrupoles are stable in a narrow region. The results uncover the existence of much richer families of stable binary QDs in comparison to states with identical components.