Yes, E is photon energy.  And, I cannot define negative energy of a photon. But, according to Jellison and Modine (Appl. Phys. Lett. 69, 371, 1996) the Forouhi-Bloomer (FB) dispersion equations violate time reversal symmetry because k(E) is not equal to -k(-E) in FB equations. However, I cannot find a specific scientific reference where the extinction coefficient, k, is explicitly associated with time reversal symmetry. Can you help?

It is well known that complex character of refraction index is resulted from solution of Maxwell wave equation  for electromagnetic waves. Like mass, charge and some  other   index refraction  is nondynamical parameter and therefore it dose not  change sign at operation of time reversal. If you change  formally  both signs  it could be connected with reversal direction of wave propagation and amplification of wave instead damping. Kramers -Kronig integral relation  in positive frequency region on the base of this complex form gave us useful  method  to connect   real and  imagine  parts of IR and showed  that connection between reason and results exists in nature. Some similarity to time reversal effect we  can meet in nonlinear optics. It is effect of wave front reversal or phase reversal at  stimulated Brilluin light  scattering to backward and  some other effect that You can find  in VIKI. So good  lack in search this  interesting question.