  We systematically construct all the tetraquark currents of $J^{PC} = 1^{++}$
with the quark configurations $[cq][\bar c \bar q]$, $[\bar c q][\bar q c]$,
and $[\bar c c][\bar q q]$ ($q=u/d$). Their relations are derived using the
Fierz rearrangement of the Dirac and color indices, through which we study
decay properties of the $X(3872)$ under both the compact tetraquark and
hadronic molecule interpretations. We propose to search for the $X(3872)
\rightarrow \chi_{c0} \pi$, $\eta_c \pi \pi$, and $\chi_{c1} \pi \pi$ decay
processes in particle experiments.
