Matrix Product Density Operators (MPDOs) provide an efficient tensor-network representation of mixed states in one-dimensional quantum many-body systems, often used to describe thermal states and steady states in dissipative dynamics. While MPDOs generalize Matrix Product States (MPS), which effectively describe 1D pure states and are associated with gapped ground states, the renormalization properties of MPDOs are more complex. In this work, we investigate real-space renormalization group (RG) transformations of MPDOs, modeled as circuits of local quantum channels. We impose that the renormalization flow must exactly preserve correlations between coarse-grained sites, ensuring that it is invertible through other local quantum channels. We first show that unlike MPS, which always admit well-defined isometric renormalization flows, MPDOs generally do not exhibit such exact flows. We introduce a subclass of MPDOs with well-defined renormalization flows, showing that these states possess a coalgebra structure and obey generalized symmetries described by Matrix Product Operator (MPO) algebras. Additionally, we explore the fixed points of these renormalization flows, providing insights into the classification of mixed-state quantum phases and the role of non-invertible symmetries in this subclass of MPDOs.