A reflexive formation is one in which the subject and object of the verb are the same. In English, reflexive pronouns are formed with the suffix “-self” as in “the man washed himself”. Quenya has a similar set of reflexive pronouns from with the prefix im- or in- (VT47/37) that presumably would be used the same way: i atan sove inse.

Person	Reflexive
1st Person Singular	imni “myself”
2nd Person Singular (familiar)	intye “yourself (familiar)”
2nd Person Singular (polite)	imle “yourself (polite)”
3rd Person Singular	inse “himself, herself”
3rd Person Singular (inanimate)	insa “itself”
1st Person Plural (exclusive)	imme “ourselves (exclusive)”
1st Person Plural (inclusive)	inwe “ourselves (inclusive)”
2nd Person Plural	inde “yourselves”
3rd Person Plural	inte “themselves”

This set of pronouns is from the late 1960s. In a somewhat earlier pronominal paradigm from 1964, Tolkien gave a set of reflexive suffixes: -kse along with dual -kset and plural -kser (PE17/75). These suffixes appear in the same row as the “impersonal inflections” used for subject verb agreement, and thus may require (and agree in number with) an independent subject: i atan sovikse, i atanu sovikset, i atani sovikser.

Conceptual Development: In the Qenya Lexicon of the 1910s, Tolkien gave a “reflexive suffix” ᴱQ. -ko, which he marked with a “?”, appearing beside a variant -to (QL/47). In The Qenya Verb Forms written somewhat later, Tolkien divided Qenya conjugations into Active, Passive and Medial (Reflexive) voice, and the Reflexives used the suffixes -sta or -kto, with masculine, feminine and neuter forms -nko, -kse, -kta (PE14/29). In the Early Quenya Grammar of the 1920s he said:

ending -kto (sg.), -lko (plural) for reflexive accusative, -ktor, -lkor dative (PE14/56).

All of these early reflexives seem to be part of similar paradigms with reflexives based on the suffix -k(t)o, with variants due to phonetic changes or combination with plural or gender indicators.

The 1964 reflexive suffix -kse could be the last iteration of these reflexive suffixes: -ko/-to >> -sta/-kto >> -kto >> -kse. If it is part of this continuum, perhaps -kse = k + se, with k being the ancient reflexive marker (hat tip to Shihali for suggesting this). It is not clear whether -kse could coexist with the im-/in- reflexive pronouns, or were replaced by them.

Neo-Quenya: Most Neo-Quenya writers stick with the late 1960s independent reflexive pronouns in im-, in-. It is conceivable that the suffix -kse (-xe) could coexist with these pronouns, and Thorsten Renk went so far as to suggest (Quetin i Lambe Eldaiva p. 48) that it was a general reflexive marker that could be combined with pronominal suffixes, as in sovixenye “I wash myself” (vs. sovinye imni). Personally I would stick with the better known independent reflexive pronouns, however.