  In the past five years, the number of known double neutron stars (DNS) in the
Milky Way has roughly doubled. We argue that the observed sample can be split
into three distinct sub-populations based on their orbital characteristics: (i)
short-period, low-eccentricity binaries; (ii) wide binaries; and (iii)
short-period, high-eccentricity binaries. These sub-populations also exhibit
distinct spin period and spindown rate properties. We focus on sub-population
(iii), which contains the Hulse-Taylor binary. Contrary to previous analysis,
we demonstrate that, if they are the product of primordial binary evolution,
the $P_{\rm orb}$ and $e$ distribution of these systems requires that the
second-born NSs must have been formed with small natal kicks ($\lesssim$25 km
s$^{-1}$) and have pre-SN masses narrowly distributed around 3.2 M$_{\odot}$.
These constraints challenge binary evolution theory and further predict closely
aligned spin and orbital axes, inconsistent with the Hulse-Taylor binary's
measured spin-orbit misalignment angle of $\approx$20$^{\circ}$. Motivated by
the similarity of these DNSs to B2127+11C, a DNS residing in the globular
cluster M15, we argue that this sub-population is consistent with being formed
in, and then ejected from, globular clusters. This scenario provides a pathway
for the formation and merger of DNSs in stellar environments without recent
star formation, as observed in the host galaxy population of short gamma ray
bursts and the recent detection by LIGO of a merging DNS in an old stellar
population.
