As you can tell from the poster, this is an Italian film, and perhaps the most important thing about it is that it stars the actor who plays the villain in Fistful of Dollars and A Few Dollars More, who, let's be honest, sort of blows Clint off the screen in both. He looks very different here because his hair is slicked down, but he's still got that face.
But what to say about this film? It's... strange. It's not really a thriller, although it's pitched as one. It's obviously a political commentary of sorts, but it also plays like psychodrama. And in places it's very surreal. The film begins with our "hero" showing up to what looks like his mistress's apartment: she greets him sultrily and asks "So, how are you going to kill me today?" "I'm going to slit your throat," he says a leer. Then they get into bed, she climbs on top of him and appears to be spasming in orgasm, but then falls over sideways and you see that she has, in fact, had his throat cut. Then our man showers off but appears to be being sloppy (leaving obvious prints around) and then he takes a fibre from his tie and pushes it under her fingernail. Then he gets dressed up in his flashy (cream colored) Italian suit and leaves. But on the way out of the gate to the apartments, he meets a young man whom he clearly knows. What's going on? Does he want to get caught? Then he drives to his office... and it turns out he's the Head of Homicide at the Police. At least, he was - he's just been promoted to Head of the "Political" Department and is only there to congratulate his successor. Of course, while he's there, news of the murder comes through and he goes along to view the crime scene. On the way out (again) he is buttonholed by a reporter and tells him "it was the husband". It looks like this is shaping up to be a film where an underling will doggedly track him down and he will be exposed for his arrogance and brought to justice - sort of Crime and Punishment crossed with Charade or something. But him planting the tie fibres was already a hint that that is not going to happen. In fact, his underlings do pick up plenty of evidence that it was him but instantly discount it because he is "above suspicion". Meanwhile he quickly takes to his new job and gives fiery speeches about clamping down on the rot in Roman society, (statistics are cited of the rate of pro-Stalin, pro-Mao and pro-Ho Chi Min graffiti through the years) and we see the brutal police interrogation methods.
(At one point someone being beaten up demands his lawyer, and the cops just snort and say "this isn't America".) We also see a lot of flashbacks of the affair between him and his victim. The explanation of her question at the beginning is that they've been acting out murder scenes as foreplay, and he has a ton of amateur photographs of her as corpse recreating crime scenes of his past investigative career.
She is a real piece of work - she seeks him out to begin with, and when she first encounters him he dresses only in sober black - it is because of her that he becomes a flashy dresser, and perhaps because of her that he becomes ambitious. But she also berates him and exposes the fact that he is a little boy underneath, which is what, in the end, seems to drive him to kill her (that and his jealousy: she was also having an affair with the young man he met on his way out, who is a well-known student revolutionary). The message of the film is clearly that power corrupts and the system is rotten. In the end he tries to get caught and even confesses, but nobody will arrest him because it would undermine the system. Or is that what happens? It's not clear - there are false endings and his victim appears again as if alive. What's real? Is any of this real? I couldn't tell. That's not to say I didn't enjoy it, but a huge amount of its charm is the lead actor: he completely carries the film, and it's fortunate he is so incredibly charismatic. But check it out and then explain it to me.