  A tetrahedron is the simplest shape that cannot be rotated into its mirror
image in 3D. The 4-Point Correlation Function (4PCF), which quantifies excess
clustering of quartets of galaxies over random, is the lowest-order statistic
sensitive to parity violation. Each galaxy defines one vertex of the
tetrahedron. Parity-odd modes of the 4PCF probe an imbalance between tetrahedra
and their mirror images. We measure these modes from the largest currently
available spectroscopic samples, the 280,067 Luminous Red Galaxies (LRGs) of
the Baryon Oscillation Spectroscopic Survey (BOSS) DR12 LOWZ ($\bar{z} = 0.32$)
and the 803,112 LRGS of BOSS DR12 CMASS ($\bar{z} = 0.57$). In LOWZ we find
$3.1\sigma$ evidence for a non-zero parity-odd 4PCF, and in CMASS we detect a
parity-odd 4PCF at $7.1\sigma$. Gravitational evolution alone does not produce
this effect; parity-breaking in LSS, if cosmological in origin, must stem from
the epoch of inflation. We have explored many sources of systematic error and
found none that can produce a spurious parity-odd \textit{signal} sufficient to
explain our result. Underestimation of the \textit{noise} could also lead to a
spurious detection. Our reported significances presume that the mock catalogs
used to calculate the covariance sufficiently capture the covariance of the
true data. We have performed numerous tests to explore this issue. The
odd-parity 4PCF opens a new avenue for probing new forces during the epoch of
inflation with 3D LSS; such exploration is timely given large upcoming
spectroscopic samples such as DESI and Euclid.
