We investigate the electrical switching of charge and spin transport in a topological insulator nanoconstriction in a four-terminal device. The switch of the edge channels is caused by the coupling between edge states which overlap in the constriction and by the tunneling effects at the contacts and therefore can be manipulated by tuning the applied voltages on the split gate or by geometrical etching. The switching mechanism can be conveniently studied by electron interferometry involving the measurements of the current in different configurations of the side gates, while the applied bias from the external leads can be tuned to obtain pure charge or pure spin currents (charge- and spin-bias configurations). Relevant signatures of quantum confinement effects, quantum size effects, and energy gap are evident in the Fabry-Perot physics of the device, allowing for a full characterization of the charge and spin currents. The proposed electrical switching behavior offers an efficient tool to manipulate topological edge-state transport in a controllable way.