Scalar-tensor theories of gravity provide mathematically equivalent descriptions of Einstein's gravity with a scalar field, in a conformally connected spacetime, described in terms of the Jordan frame and the Einstein frame. In this paper, we use the Jordan frame-Einstein frame correspondence to explore dual universes with contrasting cosmological evolutions. We study the mapping between Einstein and Jordan frames where the Einstein frame universe effectively describes the late-time evolution of the physical universe, driven by dark energy and non-relativistic matter. The Brans-Dicke theory of gravity is taken to be the dual scalar-tensor theory in the Jordan frame. We show that the standard Einstein frame universe, with dark energy and non-relativistic matter, always corresponds to a bouncing Jordan frame universe, if it is governed by a Brans-Dicke theory. This essentially leads to an alternative description of the late-time evolution of the physical universe, in terms of a bouncing Brans-Dicke universe in the Jordan frame. Previous studies have shown that for a bouncing Jordan frame, particularly for an early-time accelerating phase of the universe, the map between the Einstein and Jordan frames may become singular in the perturbative regime, causing the conformal correspondence to break down. In order to check whether the present bouncing model for late-time acceleration is free of such instabilities, we study the evolution of scalar metric perturbations. The Jordan frame metric perturbations are numerically solved, first via pulling the perturbations in the Einstein frame using the conformal correspondence, and then directly in the Jordan frame. The evolutions of perturbations obtained in these two cases are in a good agreement. Thus, the duality between the Einstein frame, mimicking the physical universe, and the bouncing Brans-Dicke Jordan frame, is shown to be stable against linear perturbations. An effective bouncing description of the current accelerating universe has interesting implications, for example, one may study the late-time cosmological perturbations as fluctuations in a bouncing scenario.