Spin-orbit effects appearing in topological insulators (TI) and at Rashba interfaces are currently revolutionizing how we can manipulate spins and have led to several newly discovered effects, from spin-charge interconversion and spin-orbit torques to novel magnetoresistance phenomena. In particular, a puzzling magnetoresistance has been evidenced, bilinear in electric and magnetic fields. Here, we report the observation of bilinear magnetoresistance (BMR) in strained HgTe, a prototypical TI. We show that both the amplitude and sign of this BMR can be tuned by controling, with an electric gate, the relative proportions of the opposite contributions of opposite surfaces. At magnetic fields of 1 T, the magnetoresistance is of the order of 1 % and has a larger figure of merit than previously measured TIs. We propose a theoretical model giving a quantitative account of our experimental data. This phenomenon, unique to TI, offer novel opportunities to tune the electrical response of surface states for spintronics.