The nature of dark matter remains one of the most important open questions in physics. Although dark matter effects have only been observed gravitationally, the order-one ratio between conventional matter and dark matter hints to a non-gravitational link between them. In this contribution, we search for such a link using the IceCube high-energy starting event (HESE) sample, which contains some of the highest energy neutrinos ever observed. These are dominated by a yet unidentified high-energy diffuse astrophysical component. Using these events we look for evidence of an excess due to dark matter decay or annihilation, and also for a novel signature due to high-energy neutrinos scattering with dark matter. Finally, in this contribution, we will summarize and highlight the complementarity between the obtained limits for these two scenarios.