As far as I understand you are interested in paraxial approximation because  εx and  εy are independent of coordinates in your equetions. Otherwise such approach is inaccessible. The angular momentum of light is depent on the field spatial distribution but not on the polarization of the beam. I case of Laguerre-Gaussian beam angular topological charge is defined by number "l" in your e( i l Φ). It can be positive or negative and it defines orbital angular momentum of light but not polarization. Polarization of Laguerre-Gaussian mode and  angular momentum are independent (for paraxial paraxial approximation at least). So  e( - i l Φ) and  e( + i l Φ) both can be used for E(R)  and E(L) in your case.

Thank you so much Dmytro. I was also thinking the same that angular momentum of light is dependent on the field spatial distribution but not on the polarization of the beam. But in Laser Photonics Rev. 2, 299 (2008) paper the E(R) caries the phase e( i l Φ) and E(L) caries the phase e(- i l Φ).

Concerning that article. That is a good review article. But there is no statement that orbital angular momentum depends on polarisation. For example you can see in introduction that sign of "l" (which defines orbital angular momentum)  and sign of "sigma" (which is associated with polarization or spin angular momentum) are independent. There are three definitions related to the angular momentum of light. Orbital angular momentum, spin angular momentum and total angular momentum (means summ of orbital and spin angular momentum, definition is used not often, so it would be better to explane definition if you use it). Orbital angular momentum depends on the field spatial distribution but not on the polarization.  Spin angular momentum depends on polarisation. They act on matter similar but not the same way, so you cant draw conclusions based only on the total angular momentum in what direction will rotate illuminated particle. The simple expression "angular momentum" as usual means "orbital angular momentum" (if not mentioned other) which is defined in your case by "l". And it is independent of polarization. So you can use any polarization for your Laguerre-Gaussian mode.