  In this work, we investigate additional signatures to support the tetraquark
mixing framework that has been recently proposed as a possible structure for
the two nonets, namely $a_0 (980)$, $K_0^* (800)$, $f_0 (500)$, $f_0 (980)$ in
the light nonet, $a_0 (1450)$, $K_0^* (1430)$, $f_0 (1370)$, $f_0 (1500)$ in
the heavy nonet. First, we advocate that the two nonets form the flavor nonet
approximately satisfying the Gell-Mann--Okubo mass relation. Then we reexamine
the mass ordering generated from the tetraquark nonets and show that this mass
ordering is satisfied by the two nonets although the ordering in the heavy
nonet is marginal. The marginal mass ordering however can be regarded as
another signature for tetraquarks because it can be explained partially by the
hyperfine masses calculated from the tetraquark mixing framework. The
tetraquark mixing parameters are found to be independent of isospins giving
additional support for the formation of the flavor nonets. In addition, we
discuss the other approaches like two-quark pictures or meson-meson bound
states, and their possible limitations in explaining the two nonets. As a
peculiar signature distinguished from other approaches, we investigate the
fall-apart coupling strengths into two vector mesons from our tetraquarks.
Coupling strengths into the two-vector modes are found to enhance strongly in
the heavy nonet while they are suppressed in the light nonet. The coupling
ratios, which depend on the isospin channel, are found to be huge around $\sim
15$. This trend in the two-vector modes, which is opposite to that in the
two-pseudoscalar fall-apart modes, can provide another testing ground for the
tetraquark mixing framework. Some experimental evidences related to the
phenomena are discussed particularly from the resonances belonging to the heavy
nonet.
