Disclaimer: the following is pure speculation and does not reflect any attested "rule" laid down by Tolkien.
The sources PE17:75 and VT49:48 list the "reflexive verbal inflection" suffix -xe (and its plural and dual forms). Tolkien however, does not provide any examples of the suffix "in the wild".
(I wonder whether Tolkien was influenced by the Slavic reflexive element as seen in Polish się, Czech se, Russian -sja.)
Later on, in the Ambidexter Sentence (VT49:21) he uses the reflexive pronoun immo to express "turn oneself" (querne immo), and provides a full paradigm for the reflexive pronoun. The current consensus among NeoQuenya users seems to be to go for the latter whenever the situation calls for a reflexive. The might be however, a use for the "orphaned" --xe suffix, and that is to express a medial voice.
Medial voice is characteristic of verbs in sentence where the is no explicit agent, the action just " happens" to the subject to the sentence: the snow melts (as if by itself and not melts by heat), the paint dries, the flood ebbs, love dwindles etc.) The are verbs that carry this medial voice by themselves, like tiku- "thaw, melt", píka- ''lessen, dwindle". But we lack attested verbs for many medial meanings, and this is where the suffix --xe could find is home. It could be the medial counterpart of the causative --ta seen in attested verbs like envinyata- "heal" and ninquita- "whiten": so where envinyata- means "heal someone" (transitive), *envinyaxa- would mean "heal, get better, convalesce". *Parkata- would mean "dry" (transitive), and *parkaxe- or *paraxe- (if derived from the root PÁRAK) would mean "dry" (intransitive).