I define a set of conditions that the most general hierarchical Yukawa mass matrices have to satisfy so that the leading rotations in the diagonalization matrix are a pair of (2,3) and (1,2) rotations. In addition to Fritzsch structures, examples of such hierarchical structures include also matrices with (1,3) elements of the same order or even much larger than the (1,2) elements. Such matrices can be obtained in the framework of a flavor theory. To leading order, the values of the angle in the (2,3) plane (s(sub 23)) and the angle in the (1,2) plane (s(sub 12)) do not depend on the order in which they are taken when diagonalizing. We find that any of the Cabbibo-Kobayashi-Maskawa matrix parametrizations that consist of at least one (1,2) and one (2,3) rotation may be suitable. In the particular case when the s(sub 13) diagonalization angles are sufficiently small compared to the product s(sub 12)s(sub 23), two special CKM parametrizations emerge: the R(sub 12)R(sub 23)R(sub 12) parametrization follows with s(sub 23) taken before the s(sub 12) rotation, and vice versa for the R(sub 23)R(sub 12)R(sub 23) parametrization. (author). 38 refs.

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