  Detailed $\gamma$-ray spectroscopy of the exotic neon isotope $^{28}$Ne has
been performed for the first time using the one-neutron removal reaction from
$^{29}$Ne on a liquid hydrogen target at 240~MeV/nucleon. Based on an analysis
of parallel momentum distributions, a level scheme with spin-parity assignments
has been constructed for $^{28}$Ne and the negative-parity states are
identified for the first time. The measured partial cross sections and momentum
distributions reveal a significant intruder $p$-wave strength providing
evidence of the breakdown of the $N=20$ and $N=28$ shell gaps. Only a weak,
possible $f$-wave strength was observed to bound final states. Large-scale
shell-model calculations with different effective interactions do not reproduce
the large $p$-wave and small $f$-wave strength observed experimentally,
indicating an ongoing challenge for a complete theoretical description of the
transition into the island of inversion along the Ne isotopic chain.
