In the past three decades, multiple studies revealed that congenital blindness is associated with functional and structural reorganization in early visual areas and its interaction with other neural systems. Among the most reproducible findings is the weaker connectivity between the visual and sensorimotor cortices, which in sighted individuals plays a role in eye-motor coordination. Here we demonstrate an important exception to this reorganization phenomena: we find that in congenitally blind individuals (as for normally sighted ones), spontaneous, non-controlled eye movements correlate with connectivity between visual and sensorimotor cortices. Furthermore, using time-lagged regression, we show that eye movements drive activity in the visual cortex, which subsequently drives sensorimotor activity. Remarkably, this phenomenon persists even though blind participants often exhibit unsynchronized eye movements and cannot sense or regulate their eye positions. These findings provide evidence of a preserved, "non-functional" connectivity in congenitally blind individuals and reveal a firm, hard-wired constraint on connectivity that remains immune to any reorganization phenomena produced by lack of visual experience and oculomotor control.