  We consider the Quartic Horndeski theory with torsion on a FLRW background in
the second order formalism. We show that there is a one parameter family of
Quartic Horndeski Cartan Lagrangians and all such theories only modify the
dispersion relations of the graviton and the scalar perturbation that are
usually found in the standard Horndeski theory on a torsionless spacetime. In
other words, for the theories in this class torsion does not induce new degrees
of freedom but it only modifies the propagation. This holds for first order
perturbations in spite of a kinetic mixing between the Horndeski scalar with
the torsion field in the action. We also show that for most Lagrangians within
the family of Quartic Horndeski Cartan theories the dispersion relation of the
scalar mode is radically modified. We find only one theory within the family
whose scalar mode has a regular wave-like dispersion relation.
