We show how the coupling between opposite edge states, which overlap in a constriction made of the topological insulator mercury telluride (HgTe), can be employed both for steering the charge flow into different edge modes and for controlled spin switching. Unlike in a conventional spin transistor, the switching does not rely on a tunable Rashba spin-orbit interaction, but on the energy dependence of the edge state wave functions. Based on this mechanism, and supported by extensive numerical transport calculations, we present two different ways to control spin and charge currents, depending on the local gating of the constriction, resulting in a high fidelity spin transistor.