  Recently, superconductivity with spontaneous time-reversal or parity symmetry
breaking is attracting much attention owing to its exotic properties, such as
nontrivial topology and nonreciprocal transport. Particularly fascinating
phenomena are expected when the time-reversal and parity symmetry are
simultaneously broken. This work shows that time-reversal symmetry-breaking
mixed-parity superconducting states generally exhibit an unusual asymmetric
Bogoliubov spectrum due to nonunitary interband pairing. For generic two-band
models, we derive the necessary conditions for the asymmetric Bogoliubov
spectrum. We also demonstrate that the asymmetric Bogoliubov quasiparticles
lead to the effective anapole moment of the superconducting state, which
stabilizes a nonuniform Fulde-Ferrell-Larkin-Ovchinnikov state at zero magnetic
fields. The concept of anapole order employed in nuclear physics, magnetic
materials science, strongly correlated electron systems, and optoelectronics is
extended to superconductors by this work. Our conclusions are relevant for any
multiband superconductors with competing even- and odd-parity pairing channels.
Especially, we discuss the superconductivity in UTe$_2$.
