Xi Jinping and Jorge Mario Bergoglio would seem like natural enemies. The Chinese president runs a government that squashes religious freedom, limits procreation and has a dismal human-rights record. Bergoglio — better known as Pope Francis — wants access to Xi’s many citizens in order to spread the Catholic Church’s teachings, challenge the Communist Party’s hold on dogma and even reach out to North Korea.

Yet these world leaders should really be sharing notes. The tasks facing the two men who took office just one day apart in 2013 are surprisingly similar — and not just because both lead flocks that number more than 1 billion people each, notes Stephan Richter, publisher of online magazine the Globalist.

“This is only where the stunning parallels start,” writes Richter. “China’s Communist Party and the Catholic Church share more than certain organizational characteristics, including being heavily male-dominated power structures. In addition, they both offer up ideologies or faith systems with an absolutist claim. That isn’t an easy proposition in an era when fewer and fewer people are inclined to adhere to such rigid propositions.”

When you look at the daunting challenges facing Xi and the pope, you really have to wonder if this is a matter of what Richter calls “two shepherds, two cultural revolutions.” Both have lots of housekeeping to do to shake up rigid bureaucracies that traffic more in ethical failures and corruption than their predecessors wanted to admit. Both are redoubling efforts to reduce poverty. Both face powerful and shadowy resistance to their pledged reforms. Both the Communist Party and the Vatican suffer from disillusionment and a loss of legitimacy among the masses. And, despite the best of intentions, it’s entirely unclear if either man will succeed.

Pope Francis, of course, has made headlines for saying and doing previously unthinkable things. Purging and investigating church leaders who were once thought untouchable — including those involved with the Vatican Bank — is causing shock waves. So are the pontiff’s moves to take on the church’s biggest scandals. His call for bishops around the globe to poll Catholics about the appeal of church teachings is a revolutionary step all its own.

Xi, too, seems to be trying to purge the system. He spent the first 12 months in office solidifying his power base. As Year 2 begins, though, the gloves may be coming off. The investigation into ex-security chief Zhou Yongkang and his family and cronies — which has netted an estimated $14.5 billion worth of illegitimate assets — may be the biggest corruption probe in modern history. Retired General Xu Caihou, a former vice chairman of the military commission under former President Hu Jintao, is also being investigated for corruption.

These moves suggest Xi’s anti-graft campaign could go deeper than officialdom anticipated. It’s early, and Xi could just be looking for some easy targets to buttress his good-governance bona fides — or just robbing Peter to pay Paul. But it’s hard to exaggerate the chilling effect all this is having in Beijing. Should these purges continue, Xi would have a freer hand to turn his big rhetoric of changing China’s growth model into reality.

After all, getting China’s government out of the economy requires reining in the powerful state-owned enterprises and shadow-banking interests that enrich party members. Just as a kind of mafia inside the Vatican’s “Roman Curia” bureaucracy has much to lose from Pope Francis’ shake-up, the many millionaires and billionaires China Inc. has created are facing uncomfortable scrutiny. Xi knows that ultimately he will have to stamp out the land grabs, illicit trading, insider trading and rent seeking that have tarnished Beijing’s image if he is to promote real change.

Xi also seems to understand, like the pope, that he needs to develop a better sense of his flock’s grievances and concerns. Along with his roles as president, military chief and party general secretary, Xi recently added Internet czar. While aimed at silencing dissent, the move is also about gauging public opinion. What’s the biggest threat to the Communist Party’s legitimacy — graft, slowing economic growth, pollution, lavish lifestyles of public officials or the one-child policy? Sinister as it is, obsessive monitoring of chat rooms and microblogging sites gives Xi a window into his subjects’ thinking.

The big question is how far these men will go to make opaque and excessively top-down organizations more transparent. I’ll defer to Vaticanologists on Pope Francis, but I fear there’s a limit to what Xi will do in his 10 years in office.

Even so, it’s heartening to see Xi beginning to address the official corruption eating away at China’s political soul. And if he ever needs a comrade to chew things over with, he may find one in the most unexpected of places.

William Pesek is a Bloomberg View columnist in Tokyo. Follow him on Twitter at @williampesek. Email: wpesek@bloomberg.net.

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