  In this paper we give explicit characterizations, based on the cutting and
spacer parameters, of (a) which rank-one transformations factor onto a given
finite cyclic permutation, (b) which rank-one transformations factor onto a
given odometer, and (c) which rank-one transformations are isomorphic to a
given odometer. These naturally yield characterizations of (d) which rank-one
transformations factor onto some (unspecified) finite cyclic permutation, (d')
which rank-one transformations are totally ergodic, (e) which rank-one
transformations factor onto some (unspecified) odometer, and (f) which rank-one
transformations are isomorphic to some (unspecified) odometer.
