Several massive black holes exhibit flux variability on time-scales that correspond to source sizes of the order of few Schwarzschild radii. We survey the potential of near-infrared and X-ray polarimetry to constrain physical properties of such black hole systems, namely, their spin and inclination. We have focused on a model where an orbiting hotspot is embedded in an accretion disc. A new method of searching for the time-lag between orthogonal polarization channels is developed and applied to an ensemble of hotspot models that samples a wide range of parameter space. We found that the hotspot model predicts signatures in polarized light which are in the range to be measured directly in the near future. However, our estimations are predicted upon the assumption of a Keplerian velocity distribution inside the flow where the dominant part of the magnetic field is toroidal. We also found that if the right model of the accretion flow can be chosen for each source (e.g. on the basis of magnetohydrodynamic simulations), then the black hole spin and inclination can be constrained to a small two-dimensional area in the spin–inclination space. The results of the application of the method to the available near-infrared polarimetric data of Sagittarius A* (Sgr A*) are presented. It is shown that even with the currently available data, the spin and inclination of Sgr A* can be constrained. Next generations of near-infrared and X-ray polarimeters should be able to exploit this tool.