  Active chiral viscoelastic materials exhibit elastic responses perpendicular
to the applied stresses, referred to as odd elasticity. We use a covariant
formulation of viscoelasticity combined with an entropy production analysis to
show that odd elasticity is not only present in active systems but also in
broad classes of passive chiral viscoelastic fluids. In addition, we
demonstrate that linear viscoelastic chiral solids do require activity in order
to manifest odd elastic responses. In order to model the phenomenon of passive
odd viscoelasticity we propose a chiral extension of Jeffreys model. We apply
our covariant formalism in order to derive the dispersion relations of
hydrodynamic modes and obtain clear imprints of odd viscoelastic behavior.
