  Quantum rotor represents the second simplest model after the harmonic
oscillator with profound interdisciplinary but yet unexplored implications. It
overreaches its mechanical meaning with significant consequences and promising
applications in, e.g., singular optics, super-conductive circuits with
Josephson junction or optimal pulse shaping in time frequency domains at
ultimate quantum limit. Quantification of complementarity between angular
momentum and angular variable is essential for exploitation of this canonical
pair in quantum metrology. Whereas the natural measures for position and
momentum are variances, the uncertainty associated with angular variable can be
linked to moments of inertia of the unit ring about axes passing through its
center of mass. This interpretation provides variants for saturable uncertainty
relations which can be further used in quantum metrology of the quantum rotor
and explains ambiguities in choice of possible uncertainty measures associated
with sine and cosine functions of angular variable. Special attention will be
payed to uncertainty products which are exactly or approximately minimised by
von Mises states, which play the role of squeezed states for quantum rotor and
allow optimal detection of quantum states at the ultimate limits.
