I've added what I think is an important paragraph to my "Salvation in Paul's Judaism" article about Paul's discontinuities as being intra or contra Judaism:
My own view is that Paul very much straddles the “contra” and “intra” Judaism fence depending on what part of his career one looks at, what we make of the gravity of his rhetoric, and contingent upon what social pressures he was facing at the time. Paul is intra-Judaism insofar as most of his community debates can normally be paralleled in halakhic discussions and they are often analogous to similar debates that took place among Jews in the Diaspora (Paul was not the first Jew to argue about food and circumcision and the Gentiles!). On the top of that, Paul’s rhetoric fits the sectarian context of second temple Judaism with rancorous polemics between sects and Paul never intended to set up a new religious entity. Yet Paul can also be seen to be contra Judaism in a very real sense as he seems to be willing to go where very few Jews would wish to follow by lowering the currency of Israel’s election in including Gentiles as part of the “Israel of God”. Indeed, Paul’s exegesis of Lev 18:5, his anthropological pessimism, the triadic link of law-sin-death that he makes, and attributing the law to angels are too raw and radical for most of his contemporaries to accept as “in-house” debates. In any case, Paul’s contrariety will depend entirely on which salvation scheme in Judaism we are talking about for it seems that Paul knew several. In Rom 1:18-32, his critique of idolatry and pagan immorality mirrors the “ethical monotheism” of Philo, Wisdom of Solomon, and Sirach as Paul doubts the existence of pagan philosophers who have a soul that is on its own sojourn towards God and the heavenly Jerusalem. Paul evidently knows of a “covenantal nomism” whereby grace is embedded in the covenant and obedience is merely the appreciative response to maintain one’s election, yet his objection remains that covenantal grace is only efficacious in the context of covenantal obedience which is precisely what Israel lacks (Romans 2–3, 9–11). Paul responds most virulently to an “ethnocentric nomism” (Gal 2:1–3:24; Phil 3:1-9) whereby Christ is merely an add-on to the Sinaitic covenant so that Christ tops-up rather than displaces the salvific function of portions of the Torah. This effectively keeps the butterfly in the cocoon and locates salvation exclusively within the Jewish constituency. Paul strenuously objects that the gospel is the good news that pagans can be saved by becoming Jews. Paul also responds to a “sapiential nomism” (1 Cor 1:10–3:23) that I postulate as a scheme arising in Corinth that perceives in Christ and the Torah a means to wisdom, power and glory. Finally, Paul opposes an “apocalyptic mysticism” that locates salvation as something acquired through law-observance coupled with visionary ascents to heaven couched in the language of Hellenistic philosophy (Colossians).
