For me it seems very clear exactly what the sign of circumcision meant for Paul virtually explains it in Phil. 3.3:  “For it is we who are the circumcision, we who serve God by his Spirit, who boast in Christ Jesus, and who put no confidence in the flesh— “(NIV)

Paul goes on to explain what “confidence in the flesh” means in the following verses as either being (for example-“of the tribe of Benjamin”) or, doing (“as for zeal, persecuting the church”). So, in the flesh, Paul could boast about these things from a natural, fleshly perspective.

Circumcision is however a removal of a piece of flesh and given as a sign of an inward condition as in Dt. 10.6: “circumcise your hearts, therefore, and do not be stiff necked any longer.” Dt. 30.6: “The Lord will circumcise your hearts and the hearts of your descendants, so that you may love the Lord with all your heart and with all your soul, and live.” Jer. 4.4: Circumcise yourselves to the Lord, circumcise your hearts, you people of Judah and the inhabitants of Jerusalem.” These O.T. verses were given to Jews who were physically circumcised but lacked spiritual vitality since they were uncircumcised in their heart.

Back to our verse in Phil. 3.3, Paul’s boast was in Christ and not his own doing. It was the Spirit’s ability in whom Paul relied. So it seems the sign shows grace and not self effort. This lines up to Abraham’s experience since he was called graciously and believed God and later received the sign of this righteousness in circumcision. So those who rely on the Spirit’s power  instead of fleshly efforts are the true circumcision.

For a somewhat different take on the rite, please see John Piper here: http://www.desiringgod.org/messages/how-do-circumcision-and-baptism-correspond