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The optimized quantum $f$-divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi relative quasi-entropy as special cases. In this paper, we establish physically meaningful refinements of the data-processing inequality for the optimized $f$-divergence. In particular, the refinements state that the absolute difference between the optimized $f$-divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a rotated Petz recovery channel. Not only do these results lead to physically meaningful refinements of the data-processing inequality for the sandwiched Rényi relative entropy, but they also have implications for perfect reversibility (i.e., quantum sufficiency) of the optimized $f$-divergences. Along the way, we improve upon previous physically meaningful refinements of the data-processing inequality for the standard $f$-divergence, as established in recent work of Carlen and Vershynina [arXiv:1710.02409, arXiv:1710.08080]. Finally, we extend the definition of the optimized $f$-divergence, its data-processing inequality, and all of our recoverability results to the general von Neumann algebraic setting, so that all of our results can be employed in physical settings beyond those confined to the most common finite-dimensional setting of interest in quantum information theory.