Altermagnets host an array of magnetic multipoles, which are often visualized and studied in reciprocal space. In real space, the relative phase of the multipoles of the spin density around atoms determines whether a system is an altermagnet or a conventional antiferromagnet. In this study, we approach these real space multipoles in altermagnets using a combination of first principles calculations and group theory. We show that, even in collinear magnets, the local spin density is necessarily noncollinear due to spin-orbit coupling. Moreover, the noncollinear contributions often provide a more direct illustration of the magnetic multipolar character of altermagnetism than the collinear contribution, which is dominated by the dipolar term. Our first principles calculations also show that 32-poles, in addition to octupoles, can be visible in the spin density of $d$-wave altermagnets, and they must be taken into account in discussions of the macroscopic response. Finally, we elucidate the interplay between magnetism and subtle crystal structural distortions in perovskite altermagnets, which provide a useful playground for studying phase transitions between antiferromagnetic and altermagnetic phases.

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• Magnetic order
• Multipole order
• Altermagnets
• Density functional theory
• Group theory