  Landau-Yang theorem is sometimes formulated as a selection rule forbidding
two real (that is, non-virtual) photons with zero total momentum to be in the
state of the total angular momentum J=1. In this paper we discuss whether the
theorem itself and this particular formulation can be extended to a pair of two
{\em twisted} photons, which carry orbital angular momentum with respect to
their propagation direction. We point out possible sources of confusion, which
may arise both from the unusual features of twisted photons and from the fact
that usual proofs of the Landau-Yang theorem operate in the center of motion
reference frame, which, strictly speaking, exists only for plane waves. We show
with an explicit calculation that a pair of twisted photons does have a
non-zero overlap with the J=1 state. What is actually forbidden is production
of a spin-1 particle by such a photon pair, and in this formulation the
Landau-Yang theorem is rock-solid. Although both the twisted photon pair and
the spin-1 particle can exist in the J=1 state, these two systems just cannot
be coupled in a gauge-invariant and Lorentz invariant manner respecting Bose
symmetry.
