  Multipolar order in bulk crystalline solids is characterized by multipole
densities -- denoted as polarizations in this work -- that cannot be cleanly
defined using the concepts of classical electromagnetism. Here we use group
theory to overcome this difficulty and present a systematic study of electric,
magnetic and toroidal multipolar order in crystalline solids. Based on our
symmetry analysis, we identify five categories of polarized matter, each of
which is characterized by distinct features in the electronic band structure.
For example, Rashba spin splitting in electropolar bulk materials like wurtzite
represents the electric dipolarization in these materials. We also develop a
general formalism of indicators for individual multipole densities that provide
a physical interpretation and quantification of multipolar order. Our work
clarifies the relation between patterns of localized multipoles and macroscopic
multipole densities they give rise to. To illustrate the general theory, we
discuss its application to polarized variants of hexagonal lonsdaleite and
cubic diamond structures. Our work provides a general framework for classifying
and expanding current understanding of multipolar order in complex materials.
