  In this paper we study almost complex and almost para-complex Cayley
structures on six-dimensional pseudo-Riemannian spheres in the space of purely
imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that
the Cayley structures are non-integrable, their basic geometric characteristics
are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there
exist (integrable) complex structures and para-complex structures on the
pseudospheres under consideration.
