The chiral vortical effect (CVE) – an axial current driven by rotation in chiral matter – appears in systems ranging from relativistic fluids to Weyl semimetals. We present an exact quantum solution of a rotating Weyl fermion in a finite cylinder. We recover the exact current density on the rotation axis obtained by Vilenkin and show that its cross-sectional average vanishes in the thermodynamic limit, establishing that the bulk vortical response is purely a magnetization current. For spin-polarized boundary conditions, we uncover an additional effect beyond the known CVE: a robust family of chiral modes that transport axial charge, Δ Q=χ N²,Δθ/4π, under rotation by angle Δθ, where χ is the Weyl node chirality and N is the number of chiral modes. The pump is independent of temperature, Fermi level, and Weyl velocities, but depends on the UV-sensitive number N. These results clarify the quantum structure of the bulk CVE and reveal a boundary-enforced chiral spectral structure underlying vortical response in Weyl systems.