  For the Kerr naked singularity (KNS) spacetimes, we study properties of
spherical photon orbits (SPOs) confined to constant Boyer-Lindquist radius r.
Some new features of the SPOs are found, having no counterparts in the Kerr
black hole (KBH) spacetimes, especially stable orbits that could be pure
prograde/retrograde, or with turning point in the azimuthal direction. At r > 1
(r < 1) the covariant photon energy E > 0 (E < 0), at r = 1 there is E = 0. All
unstable orbits must have E > 0. It is shown that the polar SPOs can exist only
in the spacetimes with dimensionless spin a < 1.7996. Existence of closed SPOs
with vanishing total change of the azimuth is demonstrated. Classification of
the KNS and KBH spacetimes in dependence on their dimensionless spin a is
proposed, considering the properties of the SPOs. For selected types of the KNS
spacetimes, typical SPOs are constructed, including the closed paths. It is
shown that the stable SPOs intersect the equatorial plane in a region of stable
circular orbits of test particles, depending on the spin a. Relevance of this
intersection for the Keplerian accretion discs is outlined and observational
effects are estimated.
