The findings are part of a joint study put forward by the Chicago Council on Global Affairs and the Levada Analytical Center, Russia's leading independent polling firm. The organizations conducted separate opinion polls over the course of the year using a similar set of questions.

In the past couple of years, Americans' feelings toward Russia, as the chart above indicates, have sunk to levels seen only during the latter years of the Cold War, when the Chicago Council on Global Affairs began conducting its poll. Controversy surrounding Russia's aggressive moves in Ukraine and the adversarial buildup to the Sochi Winter Olympics led to the cooling of attitudes. This has lingered into an election year in which Russia has loomed throughout as a dark shadow.

Meanwhile, Levada found a similar level of ill will in Russia. The chart above shows the precipitous decline in Russian attitudes toward the United States — earlier dips in the late 1990s and 2008 reflected earlier geopolitical entanglements in the Balkans and Georgia. They reflect, the report says, "Russians’ generally critical view of US engagement abroad."

One huge contrast in perception relates to how the two countries view the danger the other poses. Only 30 percent of Americans regard Russia's territorial ambitions as a "critical threat" — compared with 38 percent who regard China's military power as such. Yet 56 percent of Russians see American ambitions abroad as a "critical threat."

Perhaps for that reason, more Russians want Kremlin policy to focus on constraining Washington's influence in the world, whereas more Americans are open to encouraging cooperation and engagement with Russia. Despite the optics of the election year — where Russia has been accused of hacking Democratic National Committee emails — U.S. Democrats surveyed favor cooperation with Moscow 12 percentage points more than their Republican counterparts, 50 percent of whom opted for cooperation. (The chart below shows the totals.)

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There were quite a few points of convergence, though. More Russians (63 percent) believed Islamic fundamentalism was a "critical threat" to their country than Americans (59 percent). And around a third of respondents in both countries considered climate change to be a critical threat. Read the full report here.