If by 'reconcile' you mean to co-exist in harmony, then I expect that would entail recognizing the legitimacy of each party on its own merits. If you mean more than that, i.e. reunification, then I would direct attention of the parties to Jn. 17:17:
"Sanctify them through Thy truth. Thy Woird is truth."
Unity on the platform of God's holy Word, the Bible would be the Scriptural basis for unification. Doctrinal differences? I believe significant ones could be easily resolved in conformity to the Bible, if the parties would not bring differing agendas to the table.
Dave
The greek word used for reconcile is rather interesting. Different forms are used and no matter how one defines reconciling, JESUS promises ALL will finally be reconciled to Him, not thru denominationalism but thru One Body joined with Him in spirit and truth.
Coloss 1:20 and thru Him to reconcile/
apokatallaxai <604> (5658) the all
into Him, peace-making thru the blood of the staruos of Him thru Him whether the upon the Land whether the in the heavens
Textus Rec.) Colossians 1:20 kai di autou apokatallaxai ta panta
eiV auton eirhnopoihsaV dia tou aimatoV tou staurou autou di autou eite ta epi thV ghV eite ta en toiV ouranoiV
604. apokatallasso ap-ok-at-al-las'-so from 575 and 2644; to reconcile fully:--reconcile.
575. apo apo' a primary particle; "off," i.e. away (from something near), in various senses (of place, time, or relation; literal or figurative)....
2644. katallasso kat-al-las'-so from 2596 and 236; to change mutually, i.e. (figuratively) to compound a difference:--reconcile.
2596. kata kat-ah' a primary particle; (prepositionally) down (in place or time), in varied relations (according to the case
236. allasso al-las'-so from 243; to make different:--change.