  We calculate the masses of the $QQ\bar{q}\bar{q}$ ($Q=c,b$; $q=u,d,s$)
tetraquark states with the aid of heavy diquark-antiquark symmetry (HDAS) and
the chromomagnetic interaction (CMI) model. The masses of the highest-spin
($J=2$) tetraquarks that have only the $(QQ)_{\bar{3}_c}(\bar{q}\bar{q})_{3_c}$
color structure are related with those of conventional hadrons using HDAS.
Thereafter, the masses of their partner states are determined with the mass
splittings in the CMI model. Our numerical results reveal that: (i) the
lightest $cc\bar{n}\bar{n}$ ($n=u,d$) is an $I(J^P)=0(1^+)$ state around 3929
MeV (53 MeV above the $DD^*$ threshold) and none of the double-charm
tetraquarks are stable; (ii) the stable double-bottom tetraquarks are the
lowest $0(1^+)$ $bb\bar{n}\bar{n}$ around 10488 MeV ($\approx116$ MeV below the
$BB^*$ threshold) and the lowest $1/2(1^+)$ $bb\bar{n}\bar{s}$ around 10671 MeV
($\approx20$ MeV below the $BB_s^*/B_sB^*$ threshold); and (iii) the two lowest
$bc\bar{n}\bar{n}$ tetraquarks, namely the lowest $0(0^+)$ around 7167 MeV and
the lowest $0(1^+)$ around 7223 MeV, are near-threshold states. Moreover, we
discuss the constraints on the masses of double-heavy hadrons. Specifically,
for the lowest nonstrange tetraquarks, we obtain $T_{cc}<3965$ MeV,
$T_{bb}<10627$ MeV, and $T_{bc}<7199$ MeV.
