  A natural extension of the standard $SU(2)_{\rm L} \times U(1)_{\rm Y}$ gauge
model to accommodate massive neutrinos is to introduce one Higgs triplet and
three right-handed Majorana neutrinos, leading to a $6\times 6$ neutrino mass
matrix which contains three $3\times 3$ sub-matrices $M_{\rm L}$, $M_{\rm D}$
and $M_{\rm R}$. We show that three light Majorana neutrinos (i.e., the mass
eigenstates of $\nu_e$, $\nu_\mu$ and $\nu_\tau$) are exactly massless in this
model, if and only if $M_{\rm L} = M_{\rm D} M_{\rm R}^{-1} M_{\rm D}^T$
exactly holds. This no-go theorem implies that small but non-vanishing neutrino
masses may result from a significant but incomplete cancellation between
$M_{\rm L}$ and $M_{\rm D} M_{\rm R}^{-1} M_{\rm D}^T$ terms in the Type-II
seesaw formula, provided three right-handed Majorana neutrinos are of ${\cal
O}(1)$ TeV and experimentally detectable at the LHC. We propose three simple
Type-II seesaw scenarios with the $A_4 \times U(1)_{\rm X}$ flavor symmetry to
interpret the observed neutrino mass spectrum and neutrino mixing pattern. Such
a TeV-scale neutrino model can be tested in two complementary ways: (1)
searching for possible collider signatures of lepton number violation induced
by the right-handed Majorana neutrinos and doubly-charged Higgs particles; and
(2) searching for possible consequences of unitarity violation of the $3\times
3$ neutrino mixing matrix in the future long-baseline neutrino oscillation
experiments.
