  We systematically construct all the tetraquark currents/operators of $J^{PC}
= 1^{+-}$ with the quark configurations $[cq][\bar c \bar q]$, $[\bar c q][\bar
q c]$, and $[\bar c c][\bar q q]$ ($q=u/d$), and derive their relations through
the Fierz rearrangement of the Dirac and color indices. Using the
transformations of $[qc][\bar q \bar c] \to [\bar c c][\bar q q]$ and $[\bar c
q][\bar q c]$, we study decay properties of the $Z_c(3900)$ as a compact
tetraquark state; while using the transformation of $[\bar c q][\bar q c] \to
[\bar c c][\bar q q]$, we study its decay properties as a hadronic molecular
state.
