How To Help Your Opponents Change Your Mind and Your Client's Mind
I have alot more to say about this but for the moment am simply linking you to an article at Cognitive daily demonstrating the known fact that you are far more likely to persuade another if you are making eye contact with him.And still opposing parties resist sitting in the same room with one another when attempting to settle litigation!
There is a considerable body of research showing that eye contact is a key component of social interaction. Not only are people more aroused when they are looked at directly, but if you consistently look at the person you speak to, you will have much more social influence over that person than you would if you averted your gaze.
For full article, click here.
Emphasis on the "same room" comment mine. Whether eye-to-eye contact is a good thing for you depends on the situation. Why, for example, would I want opposing counsel staring down my injured, situationally-weakened client with no negotiation skills? The other side will exert social influence over them! That won't do me or the client any good.
Indeed, sometimes I don't want to discuss the case. Sometimes either we're at the end of the road or you're not even on our road, and I'm not going to humor you and your insufficient offers and your attempt to use social influence on me. Indeed, many of my best offers come after cancelling settlement conferences before they happen.
Just something to keep in mind. Every trick you know is a trick that can be played on you and/or your client.

Thanks SO much for starting this conversation.
I totally understand counsel's desire not to BE influenced, though if you know your case and your bottom line as I assume you do, you don't have to be worried about deviating from it -- only the untutored and unprepared are in danger of taking less or paying more than their genuine bottom line.
I also understand the desire/need for client control, but your mediator should be helping -- not hindering -- that.
I AM going to write more on the uses/benefits of the joint session really soon. But let me say here that I believe joint sessions do more harm than good if the attorneys (or the mediator) are looking for an opportunity to re-state (for the umpteenth time) their legal positions. Nor should the mediator PERMIT anyone to bully any of the parties ("stare down").
Eye contact between the parties -- if orchestrated at the right time with the consent of counsel and with appropriate safeguards -- witends to demonstrate or disprove sincerity. Sincere recitations of one party's view of the facts or about miscommunications that led to the dispute, often make the difference between settling and not settling since litigants want justice -- fairness -- not just money.
The "trick" is to help the lawyers find where their bottom lines OVERLAP. It's a shame when the parties fail to find that point of overlap because it's a missed opportunity to make a mutually beneficial deal. It's also a shame when, for example, an apology or an expression of feeling can cause the parties to move close enough together to cause one or both of them to close the gap between bottom lines.
Communication between parties about their mutual interests as well as their differing ones is not meant to be a 'trick' but an occasion to find genuine common ground.
I'd love to continue hearing your thoughts and those of your readers on this topic.