Second harmonic generation (SHG) and frequency mixing of electrical signals are fundamental for a wide range of radiofrequency applications. Recently, ferroelectric field-effect transistors (FeFETs), made from ferroelectric hafnium oxide (HfO2), have demonstrated promising SHG capabilities because of their unique symmetric transfer curves. In this paper, we illustrate how this symmetry is highly sensitive to material properties by varying the thickness of the ferroelectric layer and the doping of the silicon substrate. We show that the SHG conversion gain and the spectral purity are greatly increased (up to 96%) by precisely tuning the ferroelectric polarization reversal and the quantum tunneling currents. Based on this, we propose and experimentally demonstrate the generation of the difference and of the sum of two input frequencies (frequency mixing) with a single FeFET, which we attribute to the inherently strong quadratic component of the symmetric transfer characteristics. Because of the reversible and continuous ferroelectric switching in HfO2, our approach allows for an electrical control of the energy distribution of spectral components, thus opening up new and very promising paths for frequency manipulations with simple ferroelectric devices.