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Spinel FeMn₂O₄ is known to exhibit a cation distribution of (Mn²⁺)_A(Mn³⁺Fe³⁺)_BO₄, with Mn²⁺ mainly occupying the tetrahedral (A) sites forming a diamond lattice, and Fe³⁺ and Mn³⁺ sharing the octahedral (B) sites to form a pyrochlore lattice. Powder neutron diffraction reveals a cubic-tetragonal structural transition at ~595 K due to the Jahn-Teller distortion of MnO₆ octahedra. A collinear ferrimagnetic order with antiparallel moments between A and B sites is found below ~373 K, and non-collinear ferrimagnetic order appears below ~50 K. Spin wave dispersions in both the collinear and non-collinear ferrimagnetic ordered regions have been determined via inelastic neutron scattering on a large single crystal. At 8 K we observe a spin gap of approximately 5 meV arising from single-ion anisotropy at the orbitally active Mn³⁺ site. We argue that the orbital degree of freedom of the B-site Mn³⁺ ion and its coupling to the spin and lattice degrees of freedom play a key role in the structural and magnetic properties in FeMn₂O₄.