Spin-polarized transport in ferromagnetic tunnel junctions, characterized by tunnel magnetoresistance, has already been proven to have great potential for application in the field of spintronics and in magnetic random access memories. Until recently, in such a junction the insulating barrier played only a passive role, namely to facilitate electron tunnelling between the ferromagnetic electrodes. However, new possibilities emerged when ferroelectric materials were used for the insulating barrier, as these possess a permanent dielectric polarization switchable between two stable states. Adding to the two different magnetization alignments of the electrode, four non-volatile states are therefore possible in such multiferroic tunnel junctions. Here, we show that owing to the coupling between magnetization and ferroelectric polarization at the interface between the electrode and barrier of a multiferroic tunnel junction, the spin polarization of the tunnelling electrons can be reversibly and remanently inverted by switching the ferroelectric polarization of the barrier. Selecting the spin direction of the tunnelling electrons by short electric pulses in the nanosecond range rather than by an applied magnetic field enables new possibilities for spin control in spintronic devices.