Learn English – What are the most popular gender-neutral pronouns that aren’t the same as other contemporary pronouns

This is just another version of the he/she, him/her dilemma: English lacks singular pronouns that include both genders. I like @drm65's approach to avoiding the problem. The other likely option is to specify both:

"himself or herself"

It is unbelievable how a perpetrator will cast himself or herself in the role of victim.

Or:

"him or herself"

It is unbelievable how a perpetrator will cast him or herself in the role of victim.

Update:

Another approach is to just choose a verb that isn't reflexive:

It is unbelievable how a perpetrator will play the role of victim.

That's not always possible or best, e.g. when you're trying to emphasize exactly that reflexive aspect of the issue. But play is obviously shorter and simpler than cast him or herself in the role of, so it's worth considering unless there's a good reason to use the wordier version.

Update 2:

This question and my original answer are nearly 9 years old now, and there have been some significant changes with respect to personal pronouns in the interim. There's a greater awareness now of gender neutral and non-binary pronouns. In some contexts, a phrase like him or herself that's meant to be inclusive (compared to just himself) might be insensitive. Themself seems to be gaining acceptance even if it's not yet widely used. As the obvious singular form of the commonly used themselves, themself seems like a good choice for being inclusive while still being fairly conventional.

There are a number of other gender neutral third person reflexive pronouns such as hirself, zirself, xyrself, and coself. If you're talking about a specific person and know which pronouns they favor, use them.

Perhaps by 2029 there will be a more clear-cut answer.

Old English was highly inflected and the third person singular personal pronoun had masculine, feminine and neuter forms. The neuter form was, in the nominative, hit, which became modern English it. English grammatical gender has disappeared, but we retain he and she to refer to nouns which clearly describe male and female entities, mostly people. The neuter it refers to everything else.