Washington ? The West has made NATO’s military alliance the heart of its response to Russia’s power grab in Ukraine. But we may be fighting the wrong battle: The weapons President Vladimir Putin has used in Crimea and eastern Ukraine look more like paramilitary “covert action” than conventional military force.

Putin, the former KGB officer, may in fact be taking a page out of America’s playbook during the Ronald Reagan presidency, when the Soviet empire began to unravel thanks to a relentless U.S. covert-action campaign. Rather than confront Moscow head on, Reagan nibbled at the edges, by supporting movements that destabilized Russian power in Afghanistan, Nicaragua, Angola, and finally Poland and eastern Europe.

It was a clever American strategy back then, pushing a wounded Soviet Union and opportunistically exploiting local grievances, wherever possible. And it’s an equally clever Russian approach now, offering maximum gain at minimum potential cost.

The parallel was drawn for me this week by John Maguire, a former CIA paramilitary covert-action officer, who served in the contras program in Nicaragua and later in the Middle East. “At the end of the day, Putin is a case officer,” says Maguire. “He watched what we did in the 1980s, and now he’s playing it back against us.”  

From the beginning in Crimea, Putin ran the campaign there as a “black” operation. Russian troops wore no insignia, to preserve a fig leaf of deniability. Russian officials, from Putin on down, insisted they weren’t seizing Crimea even as their forces consolidated positions. They controlled information flows, and coordinated their messaging.

Putin’s destabilization campaign in eastern Ukraine is riskier than the Crimean operation was, because the Ukrainian government has warned it will fight back if Russian troops invade. But while the world was distracted by 50,000 Russian troops conducting maneuvers just across the border, the real action was covert destabilization of major cities in eastern Ukraine. Since these cities are largely Russian-speaking, Putin could count on a base of local popular support.

Secretary of State John Kerry tried to blow the whistle Tuesday on the Russian covert operators. “It is clear that Russian special forces and agents have been the catalyst behind the chaos of the last 24 hours,” Kerry told a Senate committee. He said the Russians might use the unrest as a pretext for military intervention, but I’d guess they would stick to their tradecraft. It’s safer, and achieves the same results with what Moscow can claim is legitimacy.

If you look back at the way the United States worked with Solidarity in Poland in the 1980s, you can see why this form of clandestine activity is so powerful. The CIA’s primary ally was the Catholic Church, headed by a Polish pope, John Paul II, who believed as a matter of religious conviction that Soviet communism should be rolled back. To work with the church, the agency needed a waiver from rules that banned operations with religious organizations.

One thing Putin learned from watching the Soviet empire fall is that the most potent weapons are those that go under the radar — and are nominally legal in the countries where operations are taking place. All the nuclear might of the Soviet Union was useless against the striking workers in Poland, or hit-and-run guerrillas in Nicaragua, or mujahedeen fighters in Afghanistan.

How can America and the West fight back effectively against Putin’s tactics? The real answer must come from Ukrainians, who will have to mobilize to protect their country from foreign meddling. Kiev made a start Tuesday, sending police to clear demonstrators from buildings in Kharkhiv but failing to oust them from government offices in Donetsk. The trick for the interim government in Kiev is to fight a nonviolent counterinsurgency — keeping a unified Ukrainian population on its side as much as possible.

The Ukrainian struggle tells us that this is a different kind of war. Putin has learned the lessons of Iraq and Afghanistan, yes, but also those of Poland and East Germany. An ex-spy is calling the shots in Moscow, using a dirty-tricks manual he knows all too well.