We show that the non-minimal conformally-coupled (CC) scalar-tensor theory and the Palatini theory with kinetic coupling of the scalar to the Ricci tensor (PKC) are the same. This is demonstrated by showing that both theories coincide in the Einstein frame. Using this duality as generating technique, we construct the PKC counterpart to the BBMB black hole of the CC theory. It turns out to be not a black hole, but a regular solution. So, in a sense, the regulating property of PKC is stronger that that of CC. We also construct dual cosmological solutions to these theories. Both begin cosmological expansion from Minkowski space. The indicated duality extends to all values of the coupling constant of the first theory, as well as to the theory with an arbitrary function of the scalar and the scalar curvature without derivatives.