A twisted arrow category is an alternative name for a category of factorisations. That latter name is applied when discussing natural systems and Baues-Wirsching cohomology, whilst the name twisted arrow category is more often used in discussing Kan extensions and within the categorical literature^{1}.
The twisted arrow category $Tw(C)$ of $C$ a category is the category of elements of its hom-functor:
Unpacking the well-known explicit construction of comma objects in $\mathbf{Cat}$ as comma categories, we get that $Tw(C)$ has
objects: $f$ an arrow in $C$, and
morphisms: between $f$ and $g$ are pairs of arrows $(p,q)$ such that the following diagram commutes:
you could view then morphisms from $f$ to $g$ as factorizations of $g$ through $f$: $g=qfp$.
From the description above, $Tw(C)$ is the same as $Arr(C)$ the arrow category of $C$, but with the direction of $p$ above in the definition of morphism reversed, hence the twist.
The (opposite of the) category $Tw(C)$ can be described as the lax colimit of the diagram $C \to Cat\colon c\mapsto C/c$.
If $C$ is a partially ordered set, then $Tw(C)$ is isomorphic to the set of nonempty intervals $[a,b]=\{c\in C| a\leq c\leq \b\}$ with $a\leq b$ in C ordered by inclusion (cf. Johnstone 1999).
which at the level of objects forgets the arrows:
and keeps everything at the morphisms level.
One could say that $Tw(C)$ classifies wedges, in the sense that for any functor $F \colon C^{op} \times C \to B$,
are the same as
This can be used to give a proof of the reduction of ends to conical limits in the $\mathbf{Set}$-enriched case, and is used in the construction of ends in a derivator.
The twisted arrow category is a special case of a category of judgments in the sense of (Melliès-Zeilberger 15).
The construction generalizes to $(\infty,1)$-categories (cf. Lurie 11, sec.4.2).
L. Errington, Twisted Systems , PhD thesis Imperial College London 1999. (doi)
Peter Johnstone, A Note on Discrete Conduché Fibrations , TAC 5 no.1 (1999) pp.1-11. (pdf)
Jacob Lurie, Derived Algebraic Geometry X: Formal Moduli Problems , ms. (2011).
Saunders Mac Lane, Categories for the Working Mathematician , Springer Heidelberg 1998². (cf. exercise IX.6.3, p.227)
Paul-André Melliès, Noam Zeilberger, Isbell Duality for Refinement Types , arXiv:1501.05115 (2015).