The Zariski Cancellation problem for smooth affine surfaces asks whether two suchsurfaces whose products with the affine line are isomorphic are isomorphic themselves. Byresults of Iitaka-Fujita, the answer is positive for surfaces of non-negative Kodaira dimen-sion. By a characterization due to Miyanishi, surfaces of negative Kodaira dimension arefibered by the affine line, and by a celebrated result of Miyanishi-Sugie, the answer to theproblem is positive if one of the surfaces is the affine plane. On the other hand, exam-ples of non-isomorphicA1-fibered affine surfaces with isomorphicA1-cylinders were firstconstructed by Danielewski in 1989, and then by many other authors. All these counter-examples are essentially constructed by variants of the method employed by Danielewski,nowadays known as the ”Danielewski fiber product trick”. In this talk, I will explain thatthis method is actually more than a trick: re-interpreted in a suitable way, it providesa necessary and sufficient criterion for twoA1-fibered surfaces over a same affine basecurveCto have isomorphic relativeA1-cylinders overC. The characterization can beroughly stated as follows: two such surfaces have isomorphic relativeA1-cylinders overCif and only if their respectively relative (log)-canonical classes are equal when viewed ascertain naturally definedQ-divisors on a non-separated orbifold curve ̆CdominatingC,fully determined by the structure of the fibers of theA1-fibrations at hand. (Joint workin progress with S. Kaliman and M. Zaidenberg).