The twisted arrow category is discussed in my paper "Extension Theories for Categories", available at http://www.cwru.edu/artsci/math/wells/pub/papers.html (it has never been published).
In Chapter IX, Sec 6, Ex 3 of CWM, MacLane defines the twisted arrow category of C such that objects are arrows f : a -> b of C and arrows are pairs of morphisms of C, (l,m) : f -> g, such that g = mfl. The construction is part of a proof of the reduction of ends to limits.
I would be grateful for pointers to other occurrences of this construction in the literature.
Charles Wells, Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-7058, USA. EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893. FAX: 216 368 5163. HOME PHONE: 440 774 1926. HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html