Given a Farey-type map F with full branches in the extended Hecke group Γm±, its dual F♯ results from constructing natural extension of F, letting time go backwards, and projecting. Although numerical simulations may suggest otherwise, we show that domain is always tame, is, it contains intervals. As main technical tool construct, for every m=3,4,5,…, homeomorphism Mm simultaneously linearizes ...