A strongly isotropy irreducible homogeneous space is a connected effective homogeneous space L/H where H is a compact subgroup of a Lie group L such that the identity component H 0 of H acts irreducibly on T~H(L/H). If we assume instead that H, but not necessarily H 0, acts irreducibly on Tetc(L/H), then L/H is called an isotropy irreducible homogeneous space. These spaces are interesting because by Schur's Lemma they all admit an L-invariant Einstein metric, and because they include the irreducible Riemannian symmetric spaces as a sub-family. The nonsymmetric strongly isotropy irreducible homogeneous spaces were classified independently by Manturov [M1,2,3], Wolf [Wol], and later by Kr~mer [Kr], with some minor omissions in the first two references. In addition, in [Wol] a detailed study of the geometric properties of strongly isotropy irreducible spaces was carried out. The isotropy irreducible spaces which are not strongly isotropy irreducible have recently been classified in [WZ3]. In a different direction, we present in this paper a direct proof of a beautiful observation of Wall concerning a correspondence between the compact simply connected irreducible symmetric spaces on the one hand and the compact strongly isotropy irreducible quotients of the classical groups on the other hand (cf. the remarks added in proof in [Wol, pp. 147-148]). This correspondence has a number of exceptions: certain Grassmannians and the isotropy irreducible space SO(7)/G 2, which is diffeomorphic to ~pT. These exceptions appear at first sight to be even more mysterious than the correspondence. But our main result is that there is an explanation, using general principles, of this correspondence as well as all its exceptions. In particular, this means that the classification of the non-symmetric strongly isotropy irreducible quotients of the classical groups may be read off from that of the irreducible symmetric spaces. Consider a compact simply connected irreducible symmetric space G/K, where G is the identity component of the isometry group, and K is the isotropy group,