  The interplay of symmetry and topology in quantum many-body mixed states has
recently garnered significant interest. In a phenomenon not seen in pure
states, mixed states can exhibit average symmetries -- symmetries that act on
component states while leaving the ensemble invariant. In this work, we
systematically characterize symmetry protected topological (SPT) phases of
short-range entangled (SRE) mixed states of spin systems -- protected by both
average and exact symmetries -- by studying their pure Choi states in a doubled
Hilbert space, where the familiar notions and tools for SRE and SPT pure states
apply. This advantage of the doubled space comes with a price: extra symmetries
as well as subtleties around how hermiticity and positivity of the original
density matrix constrain the possible SPT invariants. Nevertheless, the doubled
space perspective allows us to obtain a systematic classification of
mixed-state SPT (MSPT) phases. We also investigate the robustness of MSPT
invariants under symmetric finite-depth quantum channels, the bulk-boundary
correspondence for MSPT phases, and the consequences of the MSPT invariants for
the separability of mixed states and the symmetry-protected sign problem. In
addition to MSPT phases, we study the patterns of spontaneous symmetry breaking
(SSB) of mixed states, including the phenomenon of exact-to-average SSB, and
the order parameters that detect them. Mixed state SSB is related to an
ingappability constraint on symmetric Lindbladian dynamics.
