Neo-Orthodoxy originated as a retrieval of orthodox Christianity that gained insight through critical engagement with liberalism.  What initially connected its founding figure heads Rudolf Bultmann and Karl Barth was the reframing of biblical authority for Christian theology.  Barth, Bultmann, and the prominant American voice Reinhold Neihbur share a strong attraction to the work of Soren Kierkegaard but where they each took his philosophical insights differed.  Each figure is somewhat responsible for different theological schools that developed later, but that is for another day.  Here are the identifying makers I got, let me know if they will work for early Bultmann and Barth.

– A profound sense of the radical transcendence of God, the Holy Other;
The primacy of the initiative of God in divine revelation, the action of God in self-disclosure over against Liberalism’s human experience or discovery

–  The affirmation of the incarnation of God in Jesus Christ, accentuating the cross and resurrection (without the necessity of the virginal conception)

– The authority of the Bible as the indispensable witness to God’s revelation, but ‘infallibility’ rejected historically (because of the results of historical criticism) and theologically (because of the humanity of the Bible)

– The frank acknowledgment of radical evil, doctrinally conceived as ‘original sin’;
“I am the adam of my own soul”

– The acceptance of a nondoctrinaire biblical criticism and the revolutionary findings of modern science

– A vision of reconciliation that locates the individual in society in relation to all societal structures

– All good theologians should smoke pipes (see pictures above)