My dad once described a book by John V. Taylor about the Holy Spirit and mission as "Wayward but Brilliant"! The Go-Between God is certainly both those things. There are certain things that Taylor says in this book that have no basis in Scripture but he also demonstrates the Holy Spirit's mediator's role in a clear and concise way.
Most theologians would understand mission to be God's mission (the missio Dei) and God is a missionary God. God reaches out to humanity to draw it into the perichoretic relationship. Being God, the Holy Spirit is also an agent of mission.
How can we understand this role in an integral or wholistic way? It maybe more obvious that the Holy Spirit witnesses to Christ in a kind of evangelistic way but how about social justice and care of the creation?
Firstly, I would say that the Spirit in the Old Testament is the same Spirit of the New. So the Spirit who inspired the Prophets to denounce injustice and unrighteousness is the same Spirit who leads the Church into all truth. This truth is the truth of the God of justice and the church as a reflection (icon) of the Trinity should reflect this concern for justice as well.
Secondly, I would also say that the Holy Spirit was present at creation, hovering of the waters, and the Spirit is the one who will renew the whole creation (Romans 8:25ff). So creation care and cooperation with the Spirit is also part of the Church's mission.