  We show that simply connected Riemannian homogeneous spaces of compact
semisimple Lie groups with polar isotropy actions are symmetric, generalizing
results of Fabio Podesta and the third named author. Without assuming
compactness, we give a classification of Riemannian homogeneous spaces of
semisimple Lie groups whose linear isotropy representations are polar. We show
for various such spaces that they do not have polar isotropy actions. Moreover,
we prove that Heisenberg groups and non-symmetric Damek-Ricci spaces have
non-polar isotropy actions.
