Newbie puzzled by pipes

I'm putting in a new bathtub and moving everything a short distance in a second floor bathroom. When I opened up the wall, I found that the hot and cold water pipes have a short branch to the side where they merge into a single pipe with a valve that I'm guessing let's you set the proportion coming from each pipe. That valve was currently set right in the middle. I don't know where it goes, and I can't for the life of me imagine what it's for. But maybe it explains another riddle. In a couple of my sinks and the tub, I notice that when I first turn the cold water on, I get hot water for about a minute, then cold. Is that valve letting them mix? Should I just leave it as is, since I don't know what it's for, or turn it to one extreme position, or get rid of it. What's your take?

Maybe the original intent was to use it to divert some hot water into the cold so the toilet wouldn't sweat? That's all that comes to mind for me.

You were right. I flushed the toilet and found that with the valve turned to either extreme it got hot; in the middle is cold. I still think it's likely responsible for my hot from the cold tap nuisance. Does anyone think this valve is worth it, or should I get rid of it?

I would not keep a valve behind a closed wall without an access panel. That is for sure. Besdies being a code violation, it might leak oneday and need to be replaced.

Can you post a picture?

Some places used tempering valves for sweating toilets.
I had to remove one installed on a home with a well.
It wasn't allowing enough water for the Flushmate to seal.

Normally, when working correctly, they prevent a tank from sweating.

valve

It can cause a multitude of odd symptoms and should NEVER be hidden inside the wall.

Just imagine the fun if the valve opened enough to put mostly hot water to the toilet? Can you imagine trying to trouble shoot that call?

I went on one call that had the tempering valve exposed, so far so good.
The flapper was a barely there piece of fragmented rubber that was letting water trickle through into the bowl, the "chain" was made from rusted paper clips.
This was in the Seattle area. In all my years here, I've only seen two of these on toilets. We don't have issues with sweating tanks here.
The other tempering device has plugged itself up, and I couldn't get enough water to the Flushmate for it to seal.

Now may be a good time to remove it.

years back I had a lady call screaming that her toilet was on fire. Had a hard time not outright laughing but told her I'd be right over. When I got there she was frantic. She had shut the bathroom door and from downstairs I could hear a very loud gurgling sound. When I opened the bathroom door a cloud of steam poured out. Looking at the toilet, the water in the bowel was bubbling and steaming. Seems the tempering valve had gone bad and was letting scalding hot water from her tankless coil run to the toilet. A heating company had replaced her boiler about a week previously and never bothered to install a tempering valve on the coil. 200 degree water destroyed the fill and flapper valves and melted the wax ring.