Discrete de Rham (DDR) methods provide compatible, albeit non-conforming, approximations of the continuous de Rham complex on general polytopal meshes. However, the lack of conformity introduces specific analytical challenges. In this work, we address these issues by designing conforming liftings for the DDR spaces [1]. These liftings act as right-inverses of the interpolators and serve as key tools to overcome the aforementioned difficulties. We illustrate the utility of this approach by establishing a global integration-by-parts formula. Due to the non-conformity of the discrete complex, this formula involves a residual term, which can be interpreted as a consistency error on the adjoint of the discrete exterior derivative. Using our conforming liftings, we derive an optimal bound for this error in terms of the mesh size. The analysis is conducted within the framework of polytopal exterior calculus, enabling unified proofs across all spaces and operators of the DDR complex. Furthermore, the liftings are explicitly constructed within finite element spaces on a simplicial submesh of the underlying polytopal mesh. This construction provides enhanced control over the resulting functions, particularly regarding discrete traces and inverse inequalities.