What is this called?

There are some people at my barn who turn out their horses in the arena with the saddle on, with the horses head tied to the saddle so that his head is pulled to one side. Some of them do it gently so the horse's nose just points to one side, some do it drastically so the horses nose is pointed at the girth. After the horse has been out like this for a while they hop on and ride. The whole thing looks pretty uncomfortable to me, especially when they have the horses head really cranked to one side.

What is this called and what is it's purpose?

Frankly it's called "abuse" IMO. From the way it sounds. I've heard people doing it to teach horse "to bend", but I personally think it has nothing to do with it.

I do bend my horses on ground and in saddle, and in cutting barn I go to sometime we must stretch horses really hard on both sides and under before we get into the saddle (by using the hay or grain to encourage them :) ). I'm 100% positive the owner of that barn would kill me if I'd do something like what you describe. Lol! But I've never seen anyone actually to tie the horse head to the saddle. Very dangerous too.

They used to do it at my barn too either as a way to keep the reins up in the saddle and the horses head up while they were away or as a bending method (Though, I usually hear of that being done head to tail rather than head to saddle).

I don't think it's the smartest or most effective method, but I don't think it's abuse. I have no idea what it's "called" though, sorry.

I've seen it done lots of times before. They just called "tying their head around". Usually with young horses. I was told it was to teach them to understand bit pressure. I've neverrseen the point - I do the bending work from the saddle instead. I did see one mare fall down while tied like that, so it's something I prefer not to do. I don't think it's abuse, but not something I would do personally.

Isn't this basically checking?

No, checking isn't as severe.
It's referred to a lot as "tying around", and yes people think it helps their horse's bend more. IMO it does the opposite because the extreme flexion in the vertebrae would cause inflammation leading to osteoarthritis and eventual fusing of the vertebrae.
Other potential problems consist of the horse "hitting" the bit pressure, stopping quickly, rearing and flipping over backwards. Horses can also become very resistant to handling, saddling, etc (to the point of biting and kicking) so that they are unmanageable.
To encourage a horse to bend, lateral suppling exercises such as serpentines, circles and leg yields should be used along with proper lunging techniques.

I do it with my gelding, but I tie both reins so it teaches him to lower his head to bit pressure. He lowers his head nicely and finds his release and is comfy. Mind you I don't do it very tight, as I have seen it done very tight and young horses especailly tend to FREAK. I lunge him with his head tied back, and then I hop on and lock my hands on my legs in the same position. Works like a charm.

I've used it to help teach my gelding that it's ok to flex all the way around and touch my knee with his nose when he and I hit a roadblock in his flexing... He would only go so far, then refuse to flex further... After one fifteen minute each way session of his nose touching the saddle, he figured it out. I didn't tie his head there, though, and I used a halter... I stood with him holding the lead through one of the rings on the saddle and moved with him until he stopped turning circles, then gave him the 'relief'.

He's an absolute angel to flex all the way around now and he doesn't lock his neck in one place like my mare when he doesn't want to turn or something...

I totally agree with kitten_val although I have personally had a bad experience with tying around. My friends horse would not flex to the right so the trainer decided to tie his head around. When the horse hit that bit and it didnt give he started spinning and trying to get away. About 3 minutes later he lost it flipped over backwards and smacked his head on the wall. It killed that poor gelding instantly and my friend was devestated. It was horrible. I would never tie a horse around like that with no release to many horses can feel trapped and freak out and lose all sense of self preservation. Its horrible. But then again that's just my opinion.

Bending exercises should be done in the saddle, IMO. You can feel a horses back and how they are flexing A LOT better that way, less of a risk of injury. As far as getting a horse to carry their head low, like BrokenWings said, I lunge my horse until he is tired and starts carrying his head at a naturally low level. Horses hold their heads at a fairly horizontal angle without pulling on them, if they are feeling nice, comfortable, and calm.... in my experience anyway.