The diplomatic crisis over Turkey's downing of a Russian jet is looking increasingly like a battle of wills between Vladimir Putin and Recep Tayyip Erdogan, two leaders known for their machismo and revanchist politics. Their similar leadership style once brought them closer together, but now it's exacerbating the standoff, since neither can allow himself be seen to back down.

Putin responded to Tuesday's shooting of the Russian Su-24 jet by Turkish F-16s with predictably tough words, calling the act a "stab in the back by accomplices of terrorists" and demanding an apology.

Speaking with CNN on Thursday, Erdogan shot back that "those who violated our airspace are the ones who need to apologize." He also told France 24 that Russia was not actually fighting the Islamic State, despite Putin's claim that its Su-24s are in the Syrian skies attacking the terrorist group.

In a step toward brinksmanship rather than negotiation — Erdogan has said Putin hasn't returned his call after the incident — the Russian president has doubled down on his military presence. Although Turkey's attack was widely interpreted as a signal to Russia to stop bombing Turkmen rebels near its border in northwest Syria, Russian bombers and Syrian rocket artillery began "conducting massive strikes on this area for a long period of time" as soon as the surviving Su-24 navigator was recovered, a defense ministry representative said on Thursday. At least a dozen airstrikes reportedly hit the Turkoman Mountains where the Turkmen fighting Syrian President Bashar al-Assad are concentrated.

Russia has also deployed a guided missile cruiser to the Mediterranean and S-400 anti-aircraft missiles to Latakia airbase in Syria, extending its ground-to-air range well into Turkish territory. Independent defense analyst Alexander Golts said he was concerned that Russia could shoot down a Turkish plane to "even the score."

Along with the military posturing, Moscow is threatening economic sanctions. At the same time, experts worry that Ankara could up the ante by closing the Bosphorus Strait (a waterway which runs through the middle of Turkey), which would cut off Russia's main supply line to its forces in Syria.

'It's personal for Putin and Erdogan'

But Erdogan shouldn't be surprised at Putin's cold-shoulder response, since both have endeavored to be seen as a strong leader who can stand up to other world powers and regain past glories. They both have won popularity at home through assertive foreign policy. This image means that neither can be seen to retreat, a personal factor that is compounding the risk of conflict along with the country's conflicting interests in the region, analysts say.

"For both (Erdogan and Putin), the image question is even higher than questions of state, and that's the problem. They can't agree with each other," said Leonid Isayev, an expert of Middle Eastern politics at the Higher School of Economics in Moscow. "Both have invested a lot of their reputation in the Syrian conflict. It's personal for Putin and Erdogan, and to make any concessions is unacceptable for both of them, and this plays into the hands of IS and other terrorist groups."

Pavel Felgenhauer, an opposition-minded defense analyst, was even more blunt.

"Right now they're measuring dicks, and that's very dangerous," he said. "Of course there are also more basic problems with the whole situation, but the personal aspect contributes to possibility of further escalation."

Since coming to power, the two leaders' trajectories have been eerily similar. Both have presided over periods of unprecedented economic growth while being criticized for their authoritarian tendencies, which were on display when they cracked down on street protest movements in Moscow and Istanbul.

'These powerful, brutal men, demonstratively masculine'

They have increased their influence over the media and electoral politics, personalizing power to the point that their official title has become increasingly meaningless. In fact, Putin's 2008-12 stint as prime minister to get around presidential term limits was echoed by then-prime minister Erdogan in 2014, when he became president after exhausting his party's term limits.

Putin has sometimes been accused of being a modern czar and Erdogan a sultan, and they have reportedly both built palaces fit for such a title.

Their competitive personalities and manly images are rooted partly in athletics: Putin is renowned for his love of judo and hockey, while Erdogan once played soccer semi-professionally. Their devout religious faith is also central to their image, as is a reputation as a willful, decisive leader, and Putin once called Erdogan a "man of strong character." Russia and Turkey's warming relations in recent years have been attributed to the two leaders' personal relationship.

"The alliance of Vladimir Putin and Recep Tayyip Erdogan was in large part built on psychological closeness and Putin's attempts to look at Erdogan as a mirror," analyst Stanislav Belkovsky told radio station Ekho Moskvy this week. "He thought that they were very close, these powerful, brutal men, demonstratively masculine, who yes, deepen authoritarianism in their countries, consistently go their own way, ignoring anything they want to, including accusations of corruption."

Most importantly, both Erdogan and Putin have conducted an increasingly assertive foreign policy in recent years, which critics say is a ploy to distract from problems at home. Defiance of the West has become a way to affirm their nation's strength, independence and status as a global power.

Putin has tapped into Russians' nostalgia for the superpower status they enjoyed before the fall of the Soviet Union, which he once called the "greatest geopolitical catastrophe of the century." Meanwhile, Erdogan has revived Ottoman trappings and traditions while trying to raise Turkey's status in the region and in the Sunni Islamic world. That has included support for rebels in neighboring Syria.

"It's an absolutely identical situation. We have two pretty popular leaders, authoritarian leaders, harsh leaders who have significant support, more and more each year, mostly thanks to foreign policy, not domestic," Isayev said. "Both are pretty aggressive, we see this in Turkey and Russia. It's in demand."

'Both sides are in a position where they don't have room to maneuver'

But Russia's entry into the war in Syria, an attempt to bolster its ally Assad and assert influence in a key geopolitical region, has set the Putin and Erodgan doctrines on a collision course. Erdogan's current defiance of Russia on behalf of the Turkmen rebels, who speak Turkish and live in an area of Syria that Turkey has often seen as its historic lands, echoes Putin's own actions in Ukraine: When Russia annexed Crimea in 2014, Putin couched his actions in promises to protect Russian speakers abroad and argued Crimea was a historically Russian land. He went on to back rebels in eastern Ukraine on similar grounds, once referring to the area by a name it was called under the czarist empire.

With their reputations on the line, the clash of Erdogan and Putin's power politics has led to a standoff with a strong risk of escalation. Russians are angry and calling for a strong response to perceived Turkish aggression, while Turkish nationalist media has reveled in what it sees as Erdogan's rebuke of Putin.

"Both sides are in a position where they don't have room to maneuver and can't back away," Felgenhauer said. "They'll be calling each other's bluff, and military action of some sort between Russia and Turkey seems very possible."

Analysts argued that at this point, only a third party such as the European Union, or NATO, of which Turkey is a member, could resolve the standoff. But that intermediary would have to figure out a way for Putin and Erdogan to back down without harming their image.

"It's very risky," Golts said. "Nato would have to do something to let Erdogan save face. I don't understand how it will look."

"The issue is a conflict between two absolutely uncompromising people, two harsh positions, Russia and Turkey, and we can't solve this conflict," Isayev said. "It's a dead end."