= Solution
The <helicity> of a massless state is the component of angular momentum along its momentum,
$$
\lambda=\frac{\mathbf J\mathbin\cdot\mathbf P}{|\mathbf P|}.
$$
When the translation part of the massless little group is trivial, the <Pauli-Lubanski pseudovector> obeys
$$
W^\mu=\lambda P^\mu.
$$
A rotation through angle $\theta$ about $\mathbf P$ acts by the phase $e^{-i\lambda\theta}$. On the double cover, a $4\pi$ rotation is the identity, so
$$
e^{-4\pi i\lambda}=1
\quad\Longrightarrow\quad
\lambda\in\frac12\mathbb Z.
$$
Hence $\lambda=0,\pm\frac12,\pm1,\pm\frac32,\ldots$. A parity-invariant theory pairs the $\lambda$ and $-\lambda$ representations, although one chiral massless representation need not contain both.
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