Someone, who was, I think, eager to prove, that the persons of the Trinity are equal in all respects, gave me the example of Jesus been driven into the desert by the Holy Spirit. (Mark 1:12 "The Spirit immediately drove him out into the wilderness.") The fact that the Spirit orders Jesus into the desert isn't the trump card for proving that the Trinity do not have different roles.
The Trinity is obviously a complex and controversial topic, so anything we say needs to be said carefully. Because the "Trinity" is a extra-biblical concept to describe biblical data, whatever we say about it needs to the cumulative result God's entire special revelation. It's tempting to see this verse and think, "look the roles and persons of the Trinity are interchangeable!" Just as it might be tempting to think that when Jesus talks about his God (e.g. Matthew 27:46), it means Arius, the heretic who thought Jesus was a created being, is true. Cumulatively we understand Scripture to describe God as both one and unique, and three distinct persons in perfect community. (This is undoubtedly to our sinful finite minds a paradox but not ultimately an apologetic problem.)
The gospel accounts show God incarnate fully as a human. As a man it's appropriate that Jesus' God is the Lord of Israel. Furthermore it's the Holy Spirit, not a demonic spirit that rests upon him. Mark 1:12, while showing part of the story of Jesus, where he gets driven out into the wilderness, illustrates the paradox of the Trinity at work. Jesus the man, is sent by God the Holy Spirit into the desert while still being God incarnate. The alternative explanation is a form of modalism, where the roles of the spirit and the son are interchangeable and it doesn't matter who is sent and who does the sending.