Cross and Incarnation - One view sees the Incarnation as a threat to the Cross while another view emphasizes the Incarnation and the Resurrection over the Cross.
Prayer and Academics (Faith and Reason?) - One view sees learning as a threat to the prayer life of the church while the other sees the liturgical life of the church as an obstacle to learning.
Ministry and Liturgy - One view sees a connection between the Holy Ministry and the liturgy while another view separates the two, thus separating the Holy Ministry from the sacraments.
God of promise and God of, and in, history - One view appears to support a distant God who watches from afar and except for sending His Son to save the world does not involve Himself with the world but gives it a promise for the future. God is a watchmaker with a promise. Another view sees God as participating in human nature and man participating in divine nature through the union in Word and Sacrament.
Sin and Grace - A behaviorist inclined view tends to emphasize the former at the expense of the latter while an antinomian inclined view tends to emphasize the latter at the expense of the former. This type of dichotomy seems to offer little hope of balance or unity.
These are only a few that I have heard in recent years. They are obviously over-simplified in summary fashion here for the sake of brevity. Hopefully, one can see the false dichotomies here.
Although I am aware of these debates and more I do not always get involved in them. One thing I have discovered is there is an answer to all of them and more in the liturgy. The liturgy shows unity in matters both vertical and horizontal, the reading of Scripture, the preaching, the confession of the Creed, the distribution of the Sacrament, etc.
For example, take the Creed. The Creed confesses both Incarnation and Cross. Throughout the church year these matters, and more, are addressed. There is a unity between God and man which He has created. Here is the source of the unity that is present even when our eyes are focused on false dichotomies.