  We provide a method which takes an auto-B\"acklund transformation (auto-BT)
and produces another auto-BT for a different equation. We apply the method to
the natural auto-BTs for the ABS quad equations, which gives rise to torqued
versions of ABS equations and explains the origin of each auto-BT listed in [J.
Atkinson, J. Phys. A: Math. Theor. 41 (2008) 135202]. The method is also
applied to non-natural auto-BTs for ABS equations, which yields 3D consistent
cubes which have not been found in [R. Boll, J. Nonl. Math. Phys. 18 (2011)
337--365], and to a multi-quadratic ABS* equation giving rise to a
multi-quartic equation.
