Spiral spin liquids are magnetic states whose classical ground-state manifold consists of planar incommensurate spin spirals with wave vectors lying on a continuous ring or surface in reciprocal space. The resulting subextensive degeneracy suppresses magnetic long-range order and gives rise to liquid-like behavior. Despite numerous material realizations and theoretical investigations, the structure of low-temperature spin configurations of spin-isotropic Heisenberg spiral spin liquids has remained poorly understood. Here, we classify and characterize the classical topological defects supported by these systems. We uncover a rich family of vortex types involving concerted windings of spin directions, spiral-plane normals, and wave-vector orientations, yielding a Z × Z₂ classification. Remarkably, the elementary defects are half-vortices carrying fractional 2π windings in both spin and momentum space and obey fusion rules resembling to those of Ising anyons. Large-scale classical simulations of a square-lattice spiral spin liquid reveal that these vortices are dense in the spiral-spin-liquid regime, and bind tightly below an order-by-disorder phase transition, eventually fusing to vacuum.