  We show that supersingular K3 surfaces in characteristic $p\geq5$ are related
sequences of very special correspondences. This is not enough to conclude that
they are unirational. As a byproduct, we exhibit a fibration structure on the
moduli space of rigidified K3 crystals. We also establish Shioda-Inose type
isogeny theorems for K3 surfaces with Picard rank $\rho\geq19$ in positive
characteristic.
