The Japanese title gives a better idea of what the show's about. "Kyokou Suiri" means "Invented Inference", which is what's unique about this detective-ish story. Kotoko Iwanaga investigates mysteries and prevents trouble, but she doesn't do it by uncovering the facts. She either knows that from the start (because she can talk to ghosts or spirits who were at the crime scene) or she simply doesn't care. No, Iwanaga's modus operandi is to sort out people's problems and make everything okay, which tends to involve the opposite of truth.
This series has two levels: (a) the mystery, and (b) the characters.
The mystery is interesting, but it goes really slowly. There's lots of talking. Lots and lots and lots. The show's premise means that you can't drag things out with lots of interviews with witnesses, suspects, etc. Instead, all the show has is its detectives throwing around hypotheses and discussing stuff among themselves. There's a certain amount of action, but we know that's not what's going to solve anything. Talking is this show's bedrock.
Furthermore, almost the whole season is one big story arc to stop Steel Lady Nanase. It's even worse in the original novels/manga, in which the Guardian Snake story happens much later. (The studio moved it forwards to eps.2-3, which is a great move. It's a neat summary of what the series is about, but also more importantly gives us a bit of a breathing space before the Nanase storyline takes over everything.)
I quite like all this. It's unusual and distinctive. Iwanaga's internet tactics are clever and ingenious... but they do take a long time to get resolved.
Then we have the characters.
1. KOTOKO IWANAGA = the show's star. She's a short, one-eyed, one-legged girl who's also the Goddess of Wisdom of all youkai. (Her eye and leg were the price.) She's analytic, perhaps a bit insensitive and someone you definitely wouldn't want as an enemy. She's also rich and well-spoken. At the same time, though, she's also playful and keen to get carnal with...
2. KUROU SAKURAGAWA = whose motives and thought processes are mysterious. At the start of the series, he has a girlfriend, Saki. He'll end up going out with Iwanaga, but for a definition of "going out" that to her chagrin doesn't involve sex. He's her boyfriend, but he's dead set against doing the things that traditionally go with the role. He does have a fairly extreme secret, though. This makes him a good partner for Iwanaga, but perhaps might have made him want to avoid propagating his genes? (He has a type, admittedly, which is unfortunately the opposite of Iwanaga, but he's just as horrified in ep.8 by the prospect of intimate physical proximity to Rikka.)
3. SAKI YUMIHARA = poor Saki. She's a cop, but she's scared of creepy supernatural stuff. (In fairness, this show can get pretty nasty.)
I enjoyed this show and I'd watch a second season (which has been announced), but I don't expect to rewatch it. It would have been dry if it weren't for the characters gently winding each other up. It's definitely not a love triangle, but there's still a bit of a crackle between these characters. Kurou finds Saki sexier than Iwanaga, who for her part enjoys winding people up. And Saki, of course, didn't ask to be dragged into any of this.
It's amusing, unusual and intelligent. (I enjoyed the elegant, slightly high-flown Japanese dialogue.) You could do a lot worse.