  Two semisimple algebraic groups of the same type are said to be motivic
equivalent if the motives of the associated projective homogeneous varieties of
the same type are isomorphic. We give general criteria of motivic equivalence
in terms of the so-called higher Tits p-indices of algebraic groups. These
results allow to give a complete classification of absolutely simple classical
groups up to motivic equivalence in terms of the underlying algebraic
structures. Among other applications of this classification, we deduce that
there is a bijection between the stable birational equivalence classes of
Severi-Brauer varieties of fixed dimension and the motivic equivalence classes
of projective linear groups of the same rank.
