  Interacting dark energy-dark matter models have been widely analyzed in the
literature in an attempt to find traces of new physics beyond the usual
cosmological ($\Lambda$CDM) models. Such a coupling between both dark
components is usually introduced in a phenomenological way through a flux in
the continuity equation. However, models with a Lagrangian formulation are also
possible. A class of the latter assumes a conformal/disformal coupling that
leads to a fifth force on the dark matter component, which consequently does
not follow the same geodesics as the other (baryonic, radiation, and dark
energy) matter sources. Here we analyze how the usual cosmological
singularities of the standard matter frame are seen from the dark matter one,
concluding that by choosing an appropriate coupling, dark matter observers will
see no singularities but a non-beginning, non-ending universe. By considering
two simple phenomenological models we show that such a type of coupling can fit
observational data as well as the usual $\Lambda$CDM model.
