  Teleparallel Horndeski theory offers an avenue through which to circumvent
the speed constraint of gravitational waves in an efficient manner. However,
this provides an even larger plethora of models due to the increase in action
terms. In this work we explore these models in the context of cosmological
systems. Using Noether point symmetries, we classify the dynamical systems that
emerge from Teleparallel Horndeski cosmologies. This approach is very effective
at selecting specific models in the general class of second-order Teleparallel
scalar-tensor theories, as well as for deriving exact solutions within a
cosmological context. By iterating through the Lagrangians selected through the
Noether symmetries, we solve for a number of cosmological systems which
provides new cosmological systems to be studied.
