  We introduce twisted arrow categories of operads and of algebras over
operads. Up to equivalence of categories, the simplex category $\Delta$,
Segal's category $\Gamma$, Connes cyclic category $\Lambda$, Moerdijk-Weiss
dendroidal category $\Omega$, and categories similar to graphical categories of
Hackney-Robertson-Yau are twisted arrow categories of symmetric or cyclic
operads. Twisted arrow categories of operads admit Segal presheaves and 2-Segal
presheaves, or decomposition spaces. Twisted arrow category of an operad $P$ is
the $(\infty, 1)$-localization of the corresponding category $\Omega/P$ by the
boundary preserving morphisms. Under mild assumptions, twisted arrow categories
of operads, and closely related universal enveloping categories, are
generalized Reedy. We also introduce twisted arrow operads, which are related
to Baez--Dolan plus construction.
