  The cosmological propagation of tensor perturbations is studied in the
context of parity-violating extensions of the symmetric teleparallel equivalent
of General Relativity theory. This non-Riemannian formulation allows for a
wider variety of consistent extensions than the metric formulation of gravity
theory. It is found that while many of the possible quadratic terms do not
influence the propagation of the gravitational waves, a generic modification
predicts a signature that distinguishes the left- and right-handed circular
polarizations. The parameters of such modifications can be constrained
stringently because the propagation speed of the gravitational waves is
scale-dependent and differs from the speed of light.
