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.
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 $f$.
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 def of morphism reversed, hence the twist.
From its definition as a comma category, there’s a functor (a discrete opfibration, in fact)
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 statement above is Ex. IX.6.3 in