The equivalence between the axion invariant and the S₄ symmetry indicator is established for three-dimensional S₄-symmetric axion insulators with vanishing three-dimensional Chern numbers. Starting from the Chern-Simons expression for the magnetoelectric polarizability, 2P₃=θ/π is rewritten in terms of the S₄ sewing matrix. After stable reduction to determinant-one two-band blocks, the invariant is expressed as the degree of a map from the Brillouin zone to SU(2). The degree modulo two is then evaluated from the S₄ eigenvalues at the four S₄-invariant momenta and is shown to coincide with the symmetry indicator z₂. A minimal tight-binding model verifies the correspondence between 2P₃ and z₂. The result closes a gap between the topological-field-theory description of the axion response and the topological-band-theory classification by symmetry indicators. It also extends the known response-indicator equivalence from antiunitary settings such as CₙT symmetry to the unitary, orientation-reversing rotoinversion symmetry S₄.