The 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 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.