  We study a product of null-integrated local operators $\mathcal{O}_1$ and
$\mathcal{O}_2$ on the same null plane in a CFT. Such null-integrated operators
transform like primaries in a fictitious $d-2$ dimensional CFT in the
directions transverse to the null integrals. We give a complete description of
the OPE in these transverse directions. The terms with low transverse spin are
light-ray operators with spin $J_1+J_2-1$. The terms with higher transverse
spin are primary descendants of light-ray operators with higher spins
$J_1+J_2-1+n$, constructed using special conformally-invariant differential
operators that appear precisely in the kinematics of the light-ray OPE. As an
example, the OPE between average null energy operators contains light-ray
operators with spin $3$ (as described by Hofman and Maldacena), but also novel
terms with spin $5,7,9,$ etc.. These new terms are important for describing
energy two-point correlators in non-rotationally-symmetric states, and for
computing multi-point energy correlators. We check our formulas in a
non-rotationally-symmetric energy correlator in $\mathcal{N}=4$ SYM, finding
perfect agreement.
