Currently new radio detection techniques are being explored to detect astrophysical neutrinos beyond the PeV scale interacting in polar ice. Due to the long attenuation length of radio waves in a medium, it can be expected that such instruments will also be sensitive to the radio emission of cosmic ray air showers. Furthermore, cosmic ray air showers hitting a high-altitude layer of ice will initiate an in-ice particle cascade, also leading to radio emission. We present the first results of detailed simulations of the in-ice continuation of these cosmic-ray-induced particle cascades, using a combination of the CORSIKA Monte Carlo code and the Geant4 simulation toolkit. We give an overview of the general features of such particle cascades and present a parameterization in terms of Xmax of the longitudinal and lateral particle distributions. We discuss the feasibility of observing the in-ice particle cascades, both through the detection of the Askaryan radio emission as well as by using the RADAR reflection technique. Based on these results we find that the expected signals from the continuation of in-ice cosmic-ray induced particle cascades will be very similar to neutrino signals. This means a thorough understanding of these events is necessary in the search for neutrino candidates, while it also promises an interesting in-situ natural calibration source.