  (ABRIDGED) We use a fully self-consistent evolutionary code to follow the
rotational evolution of red giants, making a comprehensive attempt to assess
the role of rotationally induced mixing in the development of abundance
anomalies in giants with a range of masses and metallicities in stellar
clusters and the field. Unlike most previous work, we do not focus on the
determination of combinations of mixing rate and depth that reproduce the data
on a particular stellar type. Instead, we concentrate on the more fundamental
problem of the simultaneous reproduction of the CNO surface patterns in both
Population I and Population II giants using the same physics and models. A
general result of all our models is that rotational mixing, although present in
small amounts, is inefficient on the lower RGB independently of any inhibiting
effect of composition barriers. Models with differentially rotating envelopes
are able to reproduce the carbon isotope data on M67 giants with initial
rotation rates adequate to their progenitors, but fail to do so for open
clusters of larger turnoff mass as well as for metal-poor giants in the field
and globular clusters. Possible solutions are discussed. Our favored scenario
is one in which the overall strength of canonical extra mixing has been
underestimated by existent derivations, but which additionally needs to be
coupled with a much lower efficiency for rotational mixing among the rapidly
rotating open cluster giants than in the slowly rotating ones in the field and
globular clusters. We hypothesize that this last requirement is provided by the
interaction between convection and rotation in the envelopes of giants, in the
sense that rapidly rotating stars would develop much shallower angular velocity
profiles in their envelopes than do slowly rotating stars.
