  An algebraic description of basic discrete symmetries (space reversal P, time
reversal T and their combination PT) is studied. Discrete subgroups of
orthogonal groups of multidimensional spaces over the fields of real and
complex numbers are considered in terms of fundamental automorphisms of
Clifford algebras. In accordance with a division ring structure, a complete
classification of automorphisms groups is established for the Clifford algebras
over the field of real numbers. The correspondence between eight double
coverings (Dabrowski groups) of the orthogonal group and eight types of the
real Clifford algebras is defined with the use of isomorphisms between the
automorphism groups and finite groups. Over the field of complex numbers there
is a correspondence between two nonisomorphic double coverings of the complex
orthogonal group and two types of complex Clifford algebras. It is shown that
these correspondences associate with a well-known Atiyah-Bott-Shapiro
periodicity. Generalized Brauer-Wall groups are introduced on the extended sets
of the Clifford algebras. The structure of the inequality between the two
Clifford-Lipschitz groups with mutually opposite signatures is elucidated. The
physically important case of the two different double coverings of the Lorentz
groups is considered in details.
