Reflexive & Reciprocal Marked on the Verb

I am collaborating on an agglutinating conlang with polypersonal agreement
(verbs agree with their Subject; with their Primary-or-Direct Object if there is
one; and sometimes with their Secondary-or-Indirect Object if there is one.)

In this 'lang the verb's agreement-markers indicate the semantic role of the
Subject and the Direct Object.

One thing it can indicate is that the Subject, and/or the Object, is both the
Agent and the Patient of the clause.

Thus, if the clause is reflexive or reciprocal, no additional marking is needed to
establish that it must be EITHER reflexive OR reciprocal.

However, this system can't tell "reflexive" apart from "reciprocal".

That is, "Jack and Jill (each) kissed themselves" and "Jack and Jill kissed each
other" have the same agreement-markers on the verb.

How do your 'langs (whether nat- or con-) handle this?
Do they just not distinguish "reflexive" from "reciprocal"?
Do they distinguish it by marking the verb with, say, a "voice"
or "version" of "reflexive" or "reciprocal" (not the same)?
Or do they mark the difference elsewhere in the clause, say by either a
reflexive pronoun or a reciprocal pronoun or both?

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