  A number of Calder\'on-Mityagin couples and relative Calder\'on-Mityagin
pairs are identified among Banach function spaces defined in terms of the least
decreasing majorant construction on the half line. The interpolation structure
of such spaces is shown to closely parallel that of the rearrangement invariant
spaces, and it is proved that a couple of these spaces is a Calder\'on-Mityagin
couple if and only if the corresponding couple of rearrangement invariant
spaces is a Calder\'on-Mityagin couple. Consequently, the class of all
interpolation spaces for any couple of spaces of this type admits a complete
description by the K-method if and only if the class of all interpolation
spaces for the corresponding couple of rearrangement invariant spaces does.
Analogous results are proved for spaces defined in terms of the level function
construction. In the main, the conclusions for both types of spaces remain
valid when Lebesgue measure on the half line is replaced by a general Borel
measure on R. However, for certain measures the class of interpolation spaces
of these new spaces may be degenerate, reducing the "if and only if" of the
main results to a single implication.
