The AAA+ ATPase spastin remodels microtubule arrays through severing and its mutation is the most common cause of hereditary spastic paraplegias (HSP). Polyglutamylation of the tubulin C-terminal tail recruits spastin to microtubules and modulates severing activity. Here, we present a ~3.2 Å resolution cryo-EM structure of the Drosophila melanogaster spastin hexamer with a polyglutamate peptide bound in its central pore. Two electropositive loops arranged in a double-helical staircase coordinate the substrate sidechains. The structure reveals how concurrent nucleotide and substrate binding organizes the conserved spastin pore loops into an ordered network that is allosterically coupled to oligomerization, and suggests how tubulin tail engagement activates spastin for microtubule disassembly. This allosteric coupling may apply generally in organizing AAA+ protein translocases into their active conformations. We show that this allosteric network is essential for severing and is a hotspot for HSP mutations.