We investigate the motion of neutral fermions and bosons in a three-dimensional magnetic quadrupole trap. Inspecting the underlying Hamiltonian a variety of symmetries is revealed which give rise to degeneracies of the resonance energies. Our numerical approach which involves the eigenvalue problem resulting from a Sturmian basis set together with the complex scaling method enables us to calculate several hundred resonance states. The distributions of the energies and decay widths of the resonances are analyzed for both spin-1/2 fermions and spin-1 bosons. We also investigate under what conditions quasibound states with long lifetimes can be achieved. An effective scalar Schrödinger equation describing such states is derived. The results are applied to the cases of the alkali-metal atoms 87Rb and 6Li trapped in a hyperfine ground state.