I have been reading up a lot on reflexives in different languages, and this inspired me to write a little on the different types of reflexives I have encountered, and some additional types.
1. Types of pronouns
1.1. Gaps
Some languages have gaps in the pronominal system where the reflexives would be expected to appear. West Germanic, for instance, now entirely lacks reflexive possessives (whereas Slavic and North Germanic do have them).
Thus, "he saw his car" can signify the subject's car being seen by the subject, or some other third person's car being seen by the subject. In Slavic and North Germanic, there is a reflexive possessive pronoun. In examples throughout this post, I may use "sy" and "sine" for this, analogous to this structure w.r.t. Swedish: min:my::sin:sy.
Another arguable gap in most IE languages is the lack of a reflexive subject, which could make sense with subclauses and such:
"he did not know that heself would ..."
There are other imaginable situations where a reflexive nominative also makes sense, such as in split ergative languages where the nominative also is the absolutive, or in languages where some type of quirky case or differential case marking sometimes has nominative forms in object positions. Also, in the bit further down about reference, we may find other reasons.
Another imaginable gap could be a gender-specific conflation. Imagine a language where masculines and feminines have a distinct reflexive form, but neuters do not, leaving
"the animal saw it"
ambiguous as to whether it's reflexive or not.
2. Person
In some languages, reflexive pronouns are person-specific, whereas in some they are entirely person-agnostic. Russian is an example of the latter, English of the former.
Thus, in Russian, reflexive arguments often are expressed by the pronoun 'sebya' in the proper case form regardless of person (although in first and second person, using the first or second person pronouns is permitted and sometimes done). In English, it's myself, yourself, himself, herself, etc.
In Swedish, first and second person use the first and second person pronouns. The semi-reflexive "själv" (obvious cognate) can follow, but is optional.
jag såg mig in the mirror = I saw myself in the mirror
jag såg mig själv i spegeln = I saw myself (emphatic) in the mirror
I would go so far as to say that "själv" no longer is properly reflexive in Swedish but rather some kind of intensifier and restrictive marker. C.f.
Jag visste att jag själv skulle bli tvungen att lösa det.
I knew that I self would have to resolve it
Han själv hade inte hunnit med det, men med hennes hjälp gick det bra.
He self had not been_able_to_do_on_time with it, but with her help went it well
He would not have been able to get it done on time by himself, but with her help it went well (or maybe "alone, he would not ...")
Otherwise, 'själv' serves the other roles 'self' serves in English, altho' sometimes in the superlative: självaste kungen/kungen själv : the King himself.
The usual reflexive in Swedish only pops up in the third person, and does not distinguish number - although 'själv' would be inflected for plural if used with a plural, and neuter when used with neuters:
han såg sig (själv) i spegeln
de såg sig (själva) i spegeln
djuret såg sig (självt) i spegeln
he/they/the animal saw him/them/itself in the mirror, -a = plural adjective/pronoun congruence, -t = neuter ditto.
The richer the congruence system on the reflexives get in a language, the more likely it feels like the reflexive/non-reflexive distinction is going to be lost and be replaced by some form of proximative/obviative-like distinction instead. Once you have gender and number and the whole shebang both on the regular third person pronouns and reflexive ones, you will very seldom need a restriction on them that force them to be reflexive or not, but rather might just care about whether there's two referents that are distinct and of somewhat different prominence in the discourse.
3. Reference
3.1 Subject only
It is not unusual for reflexive pronouns to be restricted to subjects only. Thus, sebya and its forms, as far as I can tell, only refer back to the subject. I have no idea how this works with non-nominative subjects of infinitives in Russian, but there's space for variation there.
Natural variants of this type could be absolutive-only and topic-only.
3.2 Some other kind of reference
In Swedish, the rules for the possessive reflexive are complicated, but as an acquaintance of mine would express it: 'Any NP that C-commands the phrase with the reflexive pronoun can be the possessor'.
In fact, there's a further sort of restriction where for most speakers, the regular pronouns cannot refer to the subject (whereas the reflexive ones can; the non-reflexives do seem to be able to refer to non-subjects even in positions where they can refer to non-subjects as well.)
Two examples:
de visade honom till sitt nya rum
they showed him to sy new room
Here, the new room might be theirs or his.
John visade Peter till hans nya rum
John showed Peter to his new room
Here, the new room can only be Peter's.
The c-command rule opens up a few other positions:
tanken i all sin förträfflighet
the thought in all (of) sy excellence = the thought in all its excellence
This could be any constituent - subject, object, prepositional object, etc. This position, however, does seem to permit for use of non-reflexive pronouns as well,
tanken i all dess förtäfflighet
the thought in all (of) its excellence
Finally, the Swedish reflexive can refer to an empty subject of an infinitive, and this also holds for the regular reflexive object pronoun
att lära sig känna sina gränser är viktigt
to learn to know sy boundaries is important
att vila sig är hälsosamt
to rest oneself is healthy
Beyond this, there is a nominalization in the plural, 'de sina', which signifies the closest family and friends.
3.3. Even further kinds of reference
The Swedish example above is rather complicated, but we can find examples that it does not permit. Deciding whether to permit these (or to restrict some of the ones the Swedish example includes) can give some space for a conlang to grow into a detailed project.
3.3.1 Conjunctions
"Han och sina vänner" - "He and sy friends" does not work in Swedish. It must be "han och hans vänner".
The reflexives do not work over subclause boundaries, making
"han visste att han var försenad"
he knew that he was late
mildly ambiguous as to whether it is reflexive or not. "Själv" could be added, but would sound really weird in Swedish - though more acceptable with some other verbs in the main clause and subclause. Even then, "själv" is not necessarily reflexive, as it might actually also signify that the second subject is or does something by himself.
3.3.2 Topicalization of non-subjects
One could also imagine that topics are possible candidates for reference of reflexives, in which case you might want to be able to refer to the subject as possessed by the topic, and with a reflexive possessive at that. Or maybe even in some weird situation where the topic and the subject are the same referent, but mark different roles,
himself he gave an expensive gift.
could make much more sense in some languages' logic as
him heself gave an expensive gift.
3.3.3 Quirky case
Of course, with quirky case you may have a non-subject in the nominative or a subject in a non-nominative case, and there may be restrictions depending on whether they're true quirky cases or not affecting whether they can or can't be the referent of a reflexive. It might be nice having objects in the nominative be candidates for reflexive reference (and also, naturally, blocking the use of regular reflexive possessive pronouns owning the object).
3.3.4 Subordinate structures
It is imaginable that subordinate infinitives with an agent that is distinct from the subject of a finite verb may restrict the reflexives within its scope to refer to NPs within the infinite VP + the agent, or maybe even more restrictive, such that the agent may be blocked from being the referent of the reflexive. Thus
I helped him do his homework
could, in such a language, not be
I helped him do sy homework
One could also imagine a reverse effect, where the agent is within the scope of the infinitive phrase's block and prevents external reflexives from reaching it, thus if he helps his sister with her homework, it couldn't be
he helps sy sister do ...
But if the infinitive then permits reflexive reference to the agent, this would be permissible:
he helps his sister do sy homework
The situation with subclauses is of course of some interest as well, but I will not get into detail with regards to that. Similar possibilities exist as with regards to the infinitive phrase, but with a subclause you generally do not have arguments "outside of" the scope of the subclause.
4 Other considerations
4.1. Distinct reflexives for subjects and other referents
One could imagine a language that has evolved distinct forms of 'sy' and '...self' for subjects and objects (or ergatives and absolutives, or topics and non-topics, or subjects and objects and obliques, or topics and other NPs or maybe topics or subjects vs. non-subject non-topics). This seems unlikely but not impossible. I imagine this would likely include some kind of morphological marking distinguishing the two (or more) types instead of separate stems.
4.2 Antireflexive
I have previously considered an anti-reflexive pronoun, where the regular third person pronoun is assumed to be reflexive if there is a third person subject, and any non-reflexive reading requires an explicitly non-reflexive one.
4.3 The Finnish reflexive possessive
The Finnish reflexive possessive is fascinating in that it entirely lacks independent morphemes of its own. Normally, reflexiveness in Finnish is marked by itse + case + possessive suffix, so "itselleni" = self + to + my = to myself.
However, 'itse' does not feature in reflexive possession. 'Oma' can be used for that sometimes, but is not exclusively reflexive. In meaning it is fairly close to '(one's) own', i.e. more about exclusive possession rather than reflexive such, although both meanings do exist.
Third person reflexive possession in Finnish is expressed by the possessive suffix, whenever the noun is not the subject:
Hän löysi varastetun autonsa = (s)he found her/his stolen car
Here, we actually get a slightly antireflexive construction, because if you sneak in the genitive third person pronoun, it suddenly no longer is reflexive
Hän löysi hänen varastetun autonsa = (s)he found her/his(someone else's) stolen car