Tom Edsall, a journalism professor at Columbia who writes for the New York Times' "Campaign Stops" blog, recently undertook an illuminating project. He asked a few liberals to say what they think conservatives are right about, and he asked a few conservatives to say what they thought liberals were right about. There was some consensus on each side, and some real honest praise for the other side,  although Edsall noted that liberals were more "full throated" in their cross-party praise than were conservatives. (One small confound in Edsall's analysis: he counted me as a liberal, which I was until a year or so ago. Now I'm a centrist who is full throated in his praise of liberals and conservatives, while being quite critical of the two Parties, particularly the Republican party.)

First, here is his column on What the Right gets Right, according to liberals. Here are a few of the points of praise of conservatives:

“They appreciate more instinctively the need for fiscal balance.”

“They understand people’s more innate belief in hard work and individual responsibility and see government as too often lacking that understanding.”

They recognize “the importance of material incentives in shaping behavior, and the difficulty in keeping bureaucracies under control and responsive to citizens.”

The detect threats to "moral capital" (the resources that sustain order and trust) which liberals often cannot perceive.

And here is his column on What the Left gets Right, according to conservatives.
Common themes:

Liberals recognize the real problems facing the poor, the hardships resulting from economic globalization and the socially destructive force of increasing inequality.

Liberals do not dismiss or treat as ideologically motivated scientific findings, especially the sharpening scientific consensus that human beings contribute significantly to climate change.

Liberals stand with those most in need, and believe in the inclusion of such previously marginalized groups as blacks, Hispanics, women and gays.

Bottom line: Both sides acknowledge the "yin-yang" dynamic. Both sides realize that the other side makes important points, which their own side would not otherwise notice or respond to. Just reading Edsall's two posts back to back may be a great way to prepare for civil interactions across the divide.