  For a homogeneous space $G/H$ of reductive type, we consider the tangential
homogeneous space $G_\theta/H_\theta$. In this paper, we give obstructions to
the existence of compact Clifford-Klein forms for such tangential symmetric
spaces and obtain new tangential symmetric spaces which do not admit compact
Clifford-Klein forms. As a result, in the class of irreducible classical
semisimple symmetric spaces, we have only two types of symmetric spaces which
are not proved not to admit compact Clifford-Klein forms.
  The existence problem of compact Clifford-Klein forms for homogeneous spaces
of reductive type, which was initiated by T. Kobayashi in 1980s, has been
studied by various methods but is not completely solved yet. On the other hand,
the one for tangential homogeneous spaces has been studied since 2000s and an
analogous criterion was proved by T. Kobayashi and T. Yoshino. In concrete
examples, further works are needed to verify Kobayashi-Yoshino's condition by
direct calculations. In this paper, some easy-to-check necessary
conditions(=obstructions) for the existence of compact quotients in the
tangential setting are given, and they are applied to the case of symmetric
spaces. The conditions are related to various fields of mathematics such as
associated pair of symmetric space, Calabi-Markus phenomenon, trivializability
of vector bundle (parallelizability, Pontrjagin class), Hurwitz-Radon number
and Pfister's theorem (the existence problem of common zero points of
polynomials of odd degree).
