  Although the unitarity triangles ($UTs$) carry information about the
Kobayashi-Maskawa (KM) quark mixing matrix, it explicitly contains just three
parameters which is one short to completely fix the KM matrix. It has been
shown recently, by us, that the unitarity boomerangs ($UB$) formed using two
$UTs$, with a common inner angle, can completely determine the KM matrix and,
therefore, better represents, quark mixing. Here, we study detailed properties
of the $UBs$, of which there are a total 18 possible. Among them, there is only
one which does not involve very small angles and is the ideal one for practical
uses. Although the $UBs$ have different areas, there is an invariant quantity,
for all $UBs$, which is equal to a quarter of the Jarlskog parameter $J$
squared. Hunting new physics, with a unitarity boomerang, can reveal more
information, than just using a unitarity triangle.
