Thank you all for your insight full comments. I would say that I favor the conclusion of Thayer's Greek Lexicon (which I found here: http://biblehub.com/greek/2644.htm ) Specifically that "but the passive is used also where only one ceases to be angry with another and receives him into favor; thus καταλλαγεις, received by Cyrus into favor, Xenophon, an. 1, 6, 1; καταλλάττεται πρός αὐτήν, regained her favor, Josephus, Antiquities 5, 2, 8; and, on the other hand, God is said καταλλαγῆναι τίνι, with whom he ceases to be offended, to whom he grants his favor anew, whose sins he pardons, 2 Macc. 1:5 2Macc. 7:33 2Macc. 8:29; Josephus, Antiquities 6, 7, 4 cf. 7, 8, 4 (so ἐπικαταλλάττεσθαι τίνι, Clement of Rome, 1 Cor. 48, 1 [ET]). In the N. T. God is said καταλλάσσειν ἑαυτῷ τινα, to receive one into his favor (A. V. reconcile one to himself), 2 Corinthians 5:18f (where in the added participles two arguments are adduced which prove that God has done this: first, that he does not impute to men their trespasses; second, that he has deposited the doctrine of reconciliation in the souls of the preachers of the gospel); καταλλαγῆναι τῷ Θεῷ, to be restored to the favor of God, to recover God's favor, Romans 5:10 (but see ἐχθρός, 2); καταλλάγητε τῷ Θεῷ, allow yourselves to be reconciled to God; do not oppose your return into his favor, but lay hold of that favor now offered you, 2 Corinthians 5:20."
I am not a greek scholar. What NT greek I do know is from the studies my pastor teacher has taught us over the years and the greek tools I have either found for free on the internet or have purchased like BibleWorks 7.0 and various tools from the Olive Tree Software site. That being said, one of the very few Hermeneutical principle I do recall, is that words should be translated / interpreted with regard to how they are used in context, and as you know this word for reconciliation is used a lot here by Paul.
We see in verse 18 of 2 Cor 5, that God is doing all the work, He the Father reconciled us to Himself, "who reconciled us to Himself through Christ and gave us the ministry of reconciliation" HCSB. We serve God by serving others. One of the ways we do serve God is by telling others that God has made peace (reconciled the differences between God and the world by accepting the atoning sacrifice of His Son on the behalf of the world.
Paul further explains in verse 19 that this was done during the time that His Son, our Messiah Savior hung between heaven and earth when God decided "in Christ, God was reconciling the world to Himself, not counting their trespasses against them" Notice the them in this verse has to point back to "the world". The key here is that the reconciling work of God in Christ ended with his Spiritual and Physical deaths. God's reconciling work is complete, perfect and can not be added to by any being in the world. As such, there is nothing any being in the world can do to "be reconciled" Thus Thayer's lexicon conclusion of " allow yourselves to be reconciled to God; do not oppose your return into his favor, but lay hold of that favor now offered you, " seems to be the best translation consistent with the context and as Thayer's Greek Lexicon seems to indicate is consistent with the historical (http://biblehub.com/greek/2644.htm) use of the aorist passive imperative.
In summary I think an expanded translation of 2 Cor 5:20 would be something like this "we are ambassadors for Christ, just if He were appealing to the world through us (the message of reconciliation) we implore everyone in the world - do not oppose your return into his favor, but lay hold of that favor now offered you".
As such I respectfully disagree with the statement in one of the answers provided so far that says"To be reconciled to God through Christ therefore is to accept the free gift of eternal life through faith." We have nothing to d with reconciling our selves to God, God did all that work in the past when His Son hung from the cross and was crucified on behalf of the world
Eph 2:8 reminds us "For you are saved by grace through faith, and this is not from yourselves; it is God's gift"
Note we are saved by grace in that God the offended party choose to make peace with the entire world accepting the atoning work of Jesus Christ on behalf of not some but all of the world - and the reap the benefits of what God did in Christ on the cross by believing the message of the Father about his Son " (Mat 3:7 And there came a voice from heaven: This is My beloved Son. I take delight in Him! HCSM) (Mat 12:18 Here is My Servant whom I have chosen, My beloved in whom My soul delights; I will put My Spirit on Him, and He will proclaim justice to the nations. ) ( Mat 17:5 While he was still speaking, suddenly a bright cloud covered them, and a voice from the cloud said: This is My beloved Son. I take delight in Him. Listen to Him! HCSB)