A neutral exiton is formed from an electron and a hole, so it must have integer spin. A charged exiton would be formed from an odd number of electrons and holes, so it would have half-integer spin and corresponding magnetic moment. So the energy levels for spin up vs spin down would be split in an external magnetic field.
Thank you for clarification. Another question - how can we distinguish between positively and negatively charged exitons?
I'm not too familiar with the details in this area, but I did a quick search through some articles. Identification of the charged exiton as ##X^\pm## seems to correlate entirely with whether the material is p or n-type, which you might expect since formation of, say, ##X^-## would require a large density of electrons. I don't believe the state of the art exists to allow for direct measurement of the charge, such as by studying transport properties in an applied electric field. I found http://arxiv.org/abs/1211.0072 to be quite interesting in that they used a gate voltage to change their material from p-type to n-type and could control the formation of ##X^+## or ##X^-##. You might be able to learn more by consulting some of the classic papers in the field that are cited in that paper.