  Instigated by the first $T_{cc}$ observation at the LHCb Collaboration, the
spectroscopic properties of the entire isoscalar and isovector $T_{cc}$ family
are systematically unveiled by means of multiple sorts of relativized and
nonrelativistic diquark formalisms, encompassing the Godfrey-Isgur relativized
diquark model, the modified Godfrey-Isgur relativized diquark model
incorporating the color screening effects, the nonrelativistic diquark model
with the Gaussian type hyperfine potential, and the nonrelativistic diquark
model with the Yukawa type hyperfine potential. The theoretical outcomes of
various diquark-antidiquark scenarios are inclined to categorize the
$T_{cc}(3875)^+$ structure as the exemplary candidate of the $1S$-wave
isoscalar axial-vector double-charm tetraquark state. In light of the
diquark-antidiquark configuration, this work investigates the mixing angles of
the orbitally excited isovector $T_{cc}$ states and the magic mixing angles of
the ideal heavy-light tetraquarks for the first time. As the advancement of the
experimental detection capability, these phenomenological prognostications will
effectively boost the hunting for the prospective low-lying $T_{cc}$ states in
the future.
