  We perform a systematic analysis of the PMNS matrices which arise when one
assigns the three generations of leptons to the $2\oplus 1$ representation of a
horizontal $SU_H(2)$ symmetry. This idea has been previously explored by
Kuchimanchi and Mohapatra. However, we assume $(i)$ the neutrino mass matrix
results from leptonic couplings to $SU_L(2)$ triplet scalar fields and $(ii)$
hierarchies exist amongst lepton mass matrix elements which result from
couplings to scalar fields with different $SU_H(2)$ charges. Of the sixteen
candidate PMNS matrices which result it is found that only one is both
predictive and possesses a leading order structure compatible with experimental
data. The relevant neutrino mass matrix displays the symmetry
$L_e-L_\mu-L_\tau$ to leading order and we explore the perturbations required
to produce a realistic lepton spectrum. The effective mass in neutrinoless
double beta decay is required to lie in the range
$<m>/(10^{-2}\mathrm{eV})\in[0.7,2.5]$, which is just below current
experimental bounds. $U_{e3}$ is non-zero but not uniquely determined.
