  Developing the analysis of the distribution of the particle's
position-momentum dot product, the so-called \textit{posmom} $\mathbf{x}% \cdot
\mathbf{p}$\textbf{,} to quantum states on a circular circle on two-dimensional
Cartesian coordinates, we give its posmometry (introduced recently by Y. A.
Bernard and P. M. W. Gill, Posmom: The Unobserved Observable, J. Phys. Chem.
Lett. 1\textbf{(}2010\textbf{)}1254) for eigenstates of the free motion on the
circle, i.e., $z$-axis component of the angular momentum. The posmom has two
parity symmetries, specifically, invariant under two operations $m_{x}$ and
$m_{y}$ representing mirror symmetry about $x$ and $y$ axis respectively. The
complete eigenfunction set of the posmom is then four-valued and consists of
four basic parts each of them is defined within a distinct quadrant of the
circle. The results are not only potentially experimentally testable, but also
reflect a fact that the embedding of the circle $S^{1}$ in two-dimensional flat
space $R^{2}$ is physically reasonable.
