Proton exchange within the M–H₂ moiety of (TPB)Co(H₂) (Co–H₂; TPB = B(o-C₆H₄PⁱPr₂)₃) by 2-fold rotation about the M–H₂ axis is probed through EPR/ENDOR studies and a neutron diffraction crystal structure. This complex is compared with previously studied (SiP^iPr₃)Fe(H₂) (Fe–H₂) (SiP^iPr₃ = [Si(o-C₆H₄PⁱPr₂)₃]). The g-values for Co–H₂ and Fe–H₂ show that both have the Jahn–Teller (JT)-active ²E ground state (idealized C₃ symmetry) with doubly degenerate frontier orbitals, (e)³ = [|m_L ± 2>]³ = [x² – y², xy]³, but with stronger linear vibronic coupling for Co–H₂. The observation of ¹H ENDOR signals from the Co–HD complex, ²H signals from the Co–D₂/HD complexes, but no ¹H signals from the Co–H₂ complex establishes that H₂ undergoes proton exchange at 2 K through rotation around the Co–H₂ axis, which introduces a quantum-statistical (Pauli-principle) requirement that the overall nuclear wave function be antisymmetric to exchange of identical protons (I = 1/2; Fermions), symmetric for identical deuterons (I = 1; Bosons). Analysis of the 1-D rotor problem indicates that Co–H₂ exhibits rotor-like behavior in solution because the underlying C₃ molecular symmetry combined with H₂ exchange creates a dominant 6-fold barrier to H₂ rotation. Fe–H₂ instead shows H₂ localization at 2 K because a dominant 2-fold barrier is introduced by strong Fe(3d)→ H₂(σ*) π-backbonding that becomes dependent on the H₂ orientation through quadratic JT distortion. ENDOR sensitively probes bonding along the L₂–M–E axis (E = Si for Fe–H₂; E = B for Co–H₂). Notably, the isotropic ¹H/²H hyperfine coupling to the diatomic of Co–H₂ is nearly 4-fold smaller than for Fe–H₂.