When projection shows up in the therapy between the client and the therapist, it is given a term specific to the therapy: transference. When the projection is of the therapist toward the client, it is called counter-transference. Today we will be focusing on transference, but since this is a non-clinical forum I will stick with the more generic term of "projection."
It was Sigmund Freud who first wrote about this process. He noted how everything he did or didn't do with his clients took on seemingly magical proportions, in both positive and negative ways. He learned not to personalize what was coming his way: either the negative projections in the form of anger and criticism or the positive projections in the form of admiration or love. Instead, he used these projections as a way to better understand his client. If they were angry with him, what was it teaching him about their inner world? More specifically, who was he reminding them of from their past that they might be angry with?
It can be a marvelously healing experience for a client when they become furious at me for something I have or haven't done and rather than react with defensiveness, I simply listen with open curiosity and inquiry. First, invariably I have played a role in some way: I wasn't as sensitive as I needed to be in my choice of words, I didn't understand them as fully as I needed to, or I in some way missed the mark. And while there are incidents where that's really all there is to it (Freud said "sometimes a cigar is just a cigar") more often the client's reaction to me contains valuable information about their personal inner world that is coming to the surface via this interaction. What I have found works best in these situations is that I first have to own my own failing fully, and then when I've done that it's possible to see what made my failing so painful to the client.
Positive transference works in a similar fashion. In some ways it can be likened to romantic love. I get transformed from a person with everyday features into a magical and all-knowing being who can bring personal redemption to the person's problems. I do like to think I have something to offer, by virtue of my training, my experience, and the work I've done on myself. And just as with the negative projections, it is important to own my contributions. It does the client no good if I assume a stance of false modesty and say "Oh I don't really have that much to offer" or "what I said wasn't that important." If I do that I ruin the medicine. But just as with negative projection, it is important to help the client who is having a positive projection onto me take that projection one step further and look inward and realize that whatever they are seeing positive within me comes from a positive quality within themselves -- otherwise they wouldn't be able to see it in me.
What I am describing here is a process which would ideally unfold along these basic lines: 1) projection onto therapist, either positive or negative; 2) therapist owns their own contribution to the interaction; 3) client then is freed up to see where that comes from within and to gain greater self understanding as a result. In this way the therapy room becomes a safe laboratory for the person to discover him or herself.