"Usual suspects" announced almost immediately after the March 11th Madrid bombings were (1) Basque separatists and (2) al-Qaeda. Suggested by very few is a third possible suspect, having its roots in the post-World War II "Operation Gladio."
In 1948, the U.S. Central Intelligence Agency (CIA) set up a clandestine army in Italy. This was called "Operation Gladio" (from Latin, gladius, meaning sword). Italy was torn between the left (Communists) and the right (Fascists). Operation Gladio's purpose was to overthrow the Italian government if it went Communist. 
So too is the Roman Catholic Church, headquartered in Italy, internally torn between the left (e.g. Liberation Theology) and the right (e.g. Opus Dei). Pope John Paul I, "the smiling Pope," received every issue of a magazine published by Mino Pecorelli. Pecorelli, formerly a member of a secret Masonic society called Propaganda Due (P2), charged that 100 Freemasons had infiltrated the Vatican. Pope John Paul I ordered an investigation of the Vatican Bank but soon thereafter was assassinated. 
So when considering the Roman Catholic Church one needs to be specific about which Church: the right-wing Church or the left-wing Church.
Propaganda Due (P2), according to author Robert Anton Wilson, was "probably founded by Licio Gelli sometime in the 1970s, as part of CIA's Gladio operation..."  This seems to be the Freemasonic infiltration warned of by Pecorelli, which apparently is still entrenched in the Vatican. Pecorelli also claimed that P2 was funded by the CIA. 
The connecting webs are subtle. "The P2 story caused a major scandal in Europe but received scant attention in the American media..."  The facts that P2 founder Licio Gelli was a guest at Ronald Reagan's inauguration and was a chum of George Herbert Walker Bush are tantalizing tidbits. The P2 reliance upon a "Strategy of Tension" to further their aims is more germane.
The "Strategy of Tension" was (is) "a carefully designed plan to fabricate so much leftist terrorism that the Italians would demand an authoritarian or even fascist government."  In the United States, a similar nefarious strategy was revealed by James Bamford in his book, Body Of Secrets: "...the Joint Chiefs of Staff drew up and approved plans for what may be the most corrupt plan ever created by the U.S. government. In the name of anticommunism, they proposed a secret and bloody war of terrorism against their own country in order to trick the American public..." That plan for provocative terror against their own people in order to gain their support for fighting a Big Enemy was codenamed Operation Northwoods and was conceived in the early 1960s. 
The mist-shrouded activities of true hidden power appear only in glimpses. One glimpse is the Freemasonic propensity for symbolism paired with the numeric signatures on the September 11, 2001 terror events (9/11) and the Madrid March 11, 2004 terror events occurring 911 days after the infamous "9/11." Another glimpse is the 1980 bombing of a railway station in Bologna, Italy, reportedly originated by P2.