Then, Wikipedia should define chirality for a photon. I don't know, how that should make sense. The photon (and also gluons, but that's a bad example either, because it's pretty certain that there are no free gluons to be ever seen) is the massless realization of the Poincare group based on the representation (1/2,1/2) for the Lorentz subgroup. I don't see, how you could define a chirality transformation. Of course helicity is well defined, and since the photon is massless, it's Lorentz-invariant.
To define chirality you need a representation, where something "flips" under space reflections. To that end you have to extend the proper orthochronous Poincare group ##\mathrm{SO}(1,3)^{\uparrow}## (at least) to the orthochronous Lorentz group ##\mathrm{O}(1,3)^{\uparrow}##. For spin-1/2 particles this leads to the direct sum of the two Weyl-spinor reps. to the Dirac-spinor rep. (which is reducible wrt. to the former group but irreducible wrt. the latter), ##(1/2,0) \oplus (0,1/2)##. The Dirac bispinors can always be split into the two Weyl-spinor parts, defined by the eigenvalues of the Dirac matrix ##\gamma_5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3##. Then the parity operation is realized by switching (1/2,0) to (0,1/2), i.e., it flips the chirality, and that's why this quantity is called "chirality" in the first place. For massive particles, chirality is, however, not conserved, because the mass term in the Hamiltonian mixes left- and right-handed components. For massless particle you have chiral symmetry (modulo anomalies, but that's another story). It also turns out that for massless particles chirality and helicity are the same, indeed also helicity flips under space reflections, because momentum is a polar and angular momentum an axial vector, i.e., ##\vec{J} \cdot \vec{P}## is a pseudo scalar under space reflections.
