  We discover a class of projective self-dual algebraic varieties. Namely, we
consider actions of isotropy groups of complex symmetric spaces on the
projectivized nilpotent varieties of isotropy modules. For them, we classify
all orbit closures $X$ such that $X=\check X$ where $\check X$ is the
projective dual of $X$. We give algebraic criteria of projective self-duality
for the considered orbit closures.
