Exclusive electroproduction of ω mesons on unpolarized hydrogen and deuterium targets is studied in the kinematic region of Q2>1.0 GeV2, 3.0 GeV <W< 6.3 GeV, and −t′<0.2 GeV2. Results on the angular distribution of the ω meson, including its decay products, are presented. The data were accumulated with the HERMES forward spectrometer during the 1996–2007 running period using the 27.6 GeV longitudinally polarized electron or positron beam of HERA. The determination of the virtual-photon longitudinal-to-transverse cross-section ratio reveals that a considerable part of the cross section arises from transversely polarized photons. Spin density matrix elements are presented in projections of Q2 or −t′. Violation of s-channel helicity conservation is observed for some of these elements. A sizable contribution from unnatural-parity-exchange amplitudes is found and the phase shift between those amplitudes that describe transverse ω production by longitudinal and transverse virtual photons, γ∗L→ωT and γ∗T→ωT, is determined for the first time. A hierarchy of helicity amplitudes is established, which mainly means that the unnatural-parity-exchange amplitude describing the γ∗T→ωT transition dominates over the two natural-parity-exchange amplitudes describing the γ∗L→ωL and γ∗T→ωT transitions, with the latter two being of similar magnitude. Good agreement is found between the HERMES proton data and results of a pQCD-inspired phenomenological model that includes pion-pole contributions, which are of unnatural parity.