Essential cellular processes of microtubule disassembly and protein degradation, which span lengths from tens of $\mathsf{\mu }$m to nm, are mediated by specialized molecular machines with similar hexameric structure and function. Our molecular simulations at atomistic and coarse-grained scales show that both the microtubule-severing protein spastin and the caseinolytic protease ClpY, accomplish spectacular unfolding of their diverse substrates, a microtubule lattice and dihydrofolate reductase (DHFR), by taking advantage of mechanical anisotropy in these proteins. Unfolding of wild-type DHFR requires disruption of mechanically strong $\beta$-sheet interfaces near each terminal, which yields branched pathways associated with unzipping along soft directions and shearing along strong directions. By contrast, unfolding of circular permutant DHFR variants involves single pathways due to softer mechanical interfaces near terminals, but translocation hindrance can arise from mechanical resistance of partially unfolded intermediates stabilized by $\beta$-sheets. For spastin, optimal severing action initiated by pulling on a tubulin subunit is achieved through specific orientation of the machine versus the substrate (microtubule lattice). Moreover, changes in the strength of the interactions between spastin and a microtubule filament, which can be driven by the tubulin code, lead to drastically different outcomes for the integrity of the hexameric structure of the machine.