Context . Based on the V-shape search method, two families, Athor and Zita, have been identified within the X-complex population of asteroids located in the inner main belt. The Athor family is ~3 Gyr old while the Zita family could be as old as the Solar System. Both families were found to be capable of delivering near-Earth asteroids (NEAs). Moreover, the Athor family was linked to the low-iron enstatite (EL) meteorites. Aims . The aim of our study is to characterise the spin states of the members of the Athor and Zita collisional families and test whether these members have a spin distribution consistent with a common origin from the break up of their respective family parent asteroids. Methods . To perform this test, our method is based on the well-established asteroid family evolution, which indicates that there should be a statistical predominance of retrograde-rotating asteroids on the inward side of family’s V-shape, and prograde-rotating asteroids on the outward side of family’s V-shape. To implement the method, we used photometric data from our campaign and the literature in order to reveal the spin states, and hence their rotation sense (prograde or retrograde), of the asteroids belonging to these families. We combined dense and sparse-in-time photometric data in order to construct asteroid rotational light curves; we performed the light curve inversion method to estimate the sidereal period and 3D convex shape along with the spin axis orientation in space of several family member asteroids. Results . We obtained 34 new asteroid models for Athor family members and 17 for Zita family members. Along with the literature and revised models, the Athor family contains 60% (72% considering only the family’s core) of retrograde asteroids on the inward side and, conversely, 76% (77% considering only the family’s core) of prograde asteroids on the outward side. We also found that the Zita family exhibits 80% of retrograde asteroids on the inward side. In addition, the Zita family presents an equal amount of prograde and retrograde rotators (50% each) on the outward side. However, when we applied Kernel density estimation (KDE), we also found a clear peak for prograde asteroids on the outward side, as expected from the theory. Conclusions . The spin states of these asteroids validate the existence of both families, with the Athor family exhibiting a stronger signature for the presence of retrograde-rotating and prograde-rotating asteroids on the inner and outer side of the family, respectively. Our work provides an independent confirmation and characterisation of these very old families, whose presence and characteristics offer constraints for theories and models of the Solar System’s evolution.