Is there a verb counterpart to "pair" when bringing together three things

Yes, intrine, though don't expect most people to know it.

English provides a lot of options for grouping more than two items: group, combine, match, assemble, bring together. Whereas a pair has a lot of specific associations in English (couples, card games, chromosomes, body parts), trios are rarer, and bringing three items together (rather than two or four) is rarer still.

Rarely, people do turn trio into a verb, though in a specific sense referring to playing against an opponent in a group of three. (Example from a Final Fantasy XI encyclopedia, where it is used interchangeably with solo and duo.) So verbing trio to mean combining would possibly lead to confusion.

A more established word would be intrine. It's rare; the only example in the OED is from a translation of Dante's Divine Comedy, III.xiii.84:

The Love which with them is intrined [l. 57, l'Amor che in lor s'intrea].

Here is the excerpt more fully, from the Princeton Dante Project:

For that living Light, which so streams forth

from its shining Source that it neither parts from it

nor from the Love that is intrined with them,

of its own goodness gathers its won shining,

as though it were a mirror, in nine subsistences,

and yet eternally endures as one.

So the Light, the Source, and the Love (names for the three parts of the Christian Holy Trinity) are intrined. The word is virtually preserved from Dante's own use (the reflexive verb used is s'intrea - roughly puts itself in three), a verb that Dante coined in Italian in order to represent the Trinity:

Yet, a third variety [of neologism] turns numbers into verbs, as incinquarsi, intrearsi, immillarsi, and internarsi. Again, numbers are concrete abstractions, but Dante confounds their numerical stability, by making them reflexive verbs, numbers multiplying themselves, pointing to infinity. This is particularly appropriate for the mystery of God's infinity and trinity. (B. D. Schildgen, "Dante's Neologisms in the Paradiso and the Latin Rhetorical Tradition." Dante Studies 107, 1989.)

This may be the closest to what you are looking for ("a corresponding verb for bringing together three things"), though it is both a specific and rare usage compared to pair.