I read a lot about people concerned with our "divisive" political culture in this country (the U.S.). This is particularly interesting because it is almost always used as a criticism. One almost never hears someone say that his own party is deepening the political divisions in the country; it is always the other side. In other words, the problem of divisiveness is used in a divisive way.
It may seem natural that people would see the other side as the source of the problem, but compare this to issues like military armament. It is not difficult to find people urging unilateral disarmament on their own governments, as though the military problem would disappear if one side had no way to defend itself. On the other hand, I have yet to hear someone urge his party or faction to stop using divisive language even if the other side continues to take advantage of it. Arguably this was the tack Hilary Clinton was taking in her presidential campaign when she repeatedly said, "When they go low, you go high." But it is not unifying language to point out that you are not going to stoop to the level of your opponents.
As long as we continue to batter our political opponents with accusations of insincerity, hypocrisy, and dirty tactics, the situation is not going to get any better. These traits are used as justification for insulting language, which then gives the opponents justification for ramping up their attacks, and so on ad nauseam. I am not sure that "divisiveness" itself is a serious problem, but I am certain that believing your opponents are acting in bad faith is one. Once you believe that the other side is putting forward policies only for certain political interests rather than for the good of the country, the whole basis for having a public debate disappears. Why debate with someone who isn't arguing in good faith? If their arguments are just a ruse to distract listeners from the illicit gains to be made by some special interest, there is no need to engage those arguments. Besides, it's a lot easier to win an argument by "poisoning the well" -- claiming that your opponent's motives are bad -- rather than by dealing with the issues he raises. This leads to both sides talking past each other, raising arguments that the other side never takes seriously.
If you truly care about divisiveness, then, the worst thing you can do is to make your opponents' motives into your central point. Argue against their ideas, not their motivations. After all, even if their motivations are not sincere (and almost certainly some will be just as some will not be), their arguments still deserve to be addressed on the merits. If I make a completely insincere argument that you have no answer for, it is still a strong argument. If I am just putting up a smokescreen to cover my interests, you ought to be able to blow my arguments away with a gust of wind. If the wind blows and you find my arguments still standing, I may have a point worth considering. You will sometimes see committees appoint a devil's advocate to argue the contrary position when everyone believe that a certain action is the right one. No one thinks that the devil's advocate believes his arguments outweigh the others, but is important to think of all the possible consequences before taking a decision. Otherwise, a committee, or a nation, is inclined to rush into something in the heat of the moment without giving it full thought. Think of your opponents as serving a useful purpose in highlighting the best counterarguments. Address them. And, by taking them seriously, force your opponents to defend them seriously. As you argue about the issue, you will be building a political culture and defeating divisiveness.