Recent advances in metasurfaces have shown the importance of controlling the bianisotropic response of the constituent meta-atoms for maximum efficiency wavefront transformation. Under the paradigm of a bianisotropic metasurface, full control of the local scattering properties is allowed opening new design avenues for creating reciprocal metasurfaces. Despite recent advances in the perfect transformation of both electromagnetic and acoustic plane waves, the importance of bianisotropic metasurfaces for transforming cylindrical waves is still unexplored. Motivated by the possibility of arbitrarily controlling the angular momentum of cylindrical waves, we develop a design methodology for a bianisotropic cylindrical metasurface that enables perfect transformation of cylindrical waves. This formalism is applied to the acoustic scenario and the first experimental demonstration of perfect angular momentum transformation is shown.