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- A nd 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. © 2019 IOP Publishing Ltd and Sissa Medialab.