The Debye screening mass is a fundamental parameter characterizing the screening of chromo-electric fields in a hot quark-gluon plasma. It plays an important role in thermal field theory, entering the description of screening lengths, transport phenomena, heavy-particle interactions in the early Universe, and the dimensional reduction program that connects thermal QCD to effective three-dimensional theories. In this contribution, we revisit the reduction strategy that serves to organize the large number of terms that contribute to the screening mass at the three-loop level, applying recent advances based on canonical forms of thermal (sum-) integrands. We emphasize the role of integration variable shifts, and discuss how canonical representations of thermal integrands may simplify future high-order calculations in hot QCD.