The deformed mean field of nuclei exhibits various geometrical and dynamica
l symmetries which manifest themselves as various types of rotational and d
ecay patterns. Most of the symmetry operations considered so far have been
defined for a situation wherein the angular momentum coincides with one of
the principal axes and the principal axis cranking may be invoked. New poss
ibilities arise with the observation of rotational features in weakly defor
med nuclei and now interpreted as magnetic rotational bands. More than 120
MR bands have now been identified by filtering the existing data. We presen
t a brief overview of these bands. The total angular momentum vector in suc
h bands is tilted away from the principal axes. Such a situation gives rise
to several new possibilities including breaking of chiral symmetry as disc
ussed recently by Frauendorf. We present the outcome of such symmetries and
their possible experimental verification. Some possible examples of chiral
bands are presented.