Cleaning a barrel has more old wives tales it seems than about any other aspect of firearms.

I've read and heard the gamete from doesn't matter, to chamber side in and take brush out before you pull it back.

Seems to me that if you use all bronze brushes, plastic jags, and a good carbon fiber rod, there's no way you're going to damage anything.  The barrel steel is far far harder than the bronze, so it would be like scratching glass with your fingernails.  

Given the choice, like on a semi auto pistol barrel that comes out completly, I'll brush it from the chamber.  But on things like AR's where I'd have to get a longer rod to go through the receiver end, I just go through the muzzle end.  

And do a full in / out brush.  I do make sure I tighten the brush up after a few passes, and make sure things are straight when I'm ramming it in.  

What say you?  Some reason to be super careful and only go in one side, and or not pull the brush back, or just clean away?

I would think it is to move the gunk and fouling residue going out of the end of the barrel and not pushing or pulling it back into the chamber where it would be a lot more work to clean it out. Also not going into the muzzle end to start with would minimize any chance of damaging the crown of the rifling which could impact accuracy.

Reasonable - but isn't the few bit of rifling even more important than the last bit?  I've read it's key to have the bullet properly spun right from the start, hence how some guys use very long overall length rounds to put the bullet right up to the rifling so it doesn't have far at all to go before it's starting to be spun.

For gunk, the brush is going to have as much on it if you pull it back or if you take it off, screw it back on, and run it in again.  Maybe you're pushing gunk to the end of the barrel and keeping it there where the next pass will help to push it out.

I think most people over clean guns as it is.  I think it comes from the military, we used to clean just to be given something to do even when all we did the last time we had them out was clean them.  Lots of people learn that in the military & just keep doing it & tell others it's how it's supposed to be done, or else.  I run a swab or patch w/ Hoppe's down the barrel, let it soak for a few, run a bore snake, then I spray a bit of ballistol down the bore & gas tube then run the snake through again.  Clean the BCG & that's pretty much it unless there is some gunk, dirt or debris I noticed somewhere else when field stripping.  I also have never been a big believer that you must run it from breech to muzzle or you will damage the gun, if it can handle a bullet going down the barrel at 2K-3K+ fps I don't think a swab, patch, jag, brush, snake, etc. are going to ruin it if you run it from muzzle to chamber.

I would imagine that if you are using a fixed cleaning rod it might be necessary, it you are using a rotating one, not so much.
I run it back and forth a few times then pour boiling water through the barrel followed by a dry patch.

I've been using segmented rods ever since I got my first rifle. That puts it over 15 years of using it. I still have parts of my first rod that I use on my new rods. I recently checked my rifles with a bore cam and found that other than having to clean the throat area better, nothing is damaged. No crown damage, nothing.

I always drop the rod in, and pull it through. I use various bronze brushes, nylon brushes, steel brushes, and patches. Sometimes I would get a patch stuck too and end up breaking my segmented rod hammering it out. Over the last few years, I used a chamber guide and push the rod through. Yes a segmented rod, right through the barrel. That's like committing every sin imaginable, right?

There were times when a patch would get stuck. One time I used this Hoppes elite bore snake with a thin line on it and it snapped off mid bore. It was so hard to take out yet on that rifle, no sign of any sort of damage with a bore camera. Other times I would break a segmented rod trying to hammer out a stuck patch.

I also use every type of chemical imaginable. From pastes, to ammonia based cleaners like Sweets, and the standard CLP type cleaners.

Yet not a single sign of damage that I can confirm with a bore camera. I also have a throat erosion gauge, and even in my all fire cracked rifle, it hasn't moved even half a notch. Not a single noticeable degrade in accuracy, or anything else.

The thing I always notice is no matter what, my rods are always centered. When I pull, I'm not dragging it on the bore or using the crown as a pivot point.

Now I'm not an expert by any means and maybe I'm slowly making my rifles into a smooth bore, I don't know. But from what I can tell, so long as you aren't going crazy and pushing the rod in and out like you're playing a violin at a concert, then I don't think you will do any significant damage.

I use a Dewey coated one piece rod, brass or nylon brushes, patches, CLP, and a chamber guide. Bore snakes are quick and easy especially if I didn't shoot much on a range day.

I like to compare what you are doing cleaning to what happens when you fire the gun.

When you fire the gun, you a forcing an oversized projectile squeezing it through the barrel at incredible speed.  If it is your second shot, that bullet it passing over all the carbon/powder debris etc etc that is already in the barrel and the shape of the bullet would act as a wedge really cramming that stuff against the barrel.  If you think about it, you really have to laugh when people talk about how much damage you do "over cleaning" a barrel (even though a lot of well respected firearm experts talk about it).

For a super precision rifle, I will go one way from bore to muzzle and usually unscrew the jag/brush before pulling rod back in.  

For most rifles I just pull the jag/brush back through.
I DO re-use patches.  If you think pushing a dirty patch is going to damage the barrel, see comment above about firing a second shot.

Unless you are using a rat-tail file as your cleaning rod, I think you will have a hard time doing more damage to a barrel than a fired bullet would do.

If cleaning is going to damage your barrel, and God forbid the damage firing a second bullet would do, you basically should just replace your barrel every shot...

If I have the time, I go breech-muzzle. On the 500rd deeper clean with JB bore paste, I'll work the loop jag back and forth in the first few inches, but for cleaning that out, breech-muzzle.

I don't have data, but it seems like it should reduce the chance of wearing the crown, and keep the clean bore shots closer to the main group.

Come on. All you military guys should have figured this out within your first 6 months. Take 3 sections of rod and the brush and eyelet. drop the oil soaked patch in from the chamber end and pull through. Let it sit for a few minutes as you clean your bolt. Then using the brush pull it through a few times and wet patch again. Repeat until it's coming out light gray. You pull it to make it more efficient than anything else. Use the 1" square patch and it will never get stuck. Use the T handle only for the chamber brush.

Exactly.  Why push the crud from the barrel into the receiver/barrel lug area?

Pulling can keep the rod straight (it's in tension) if you go slow and pay attention.  Straight and centered keeps it off the lands.

Pushing will force the rod to bend (it's being compressed, or under compressive force) and rub the lands.

Believe it hurts or not, you can rub, or not rub.

I've used Hoppe's #9 for over 40 years.  But, a few months ago I bought some stuff called Bore Tech Eliminator (thanks to my buddy who told me about it) and I go through far less patches getting the crud out of the barrel so the barrel is getting cleaner faster than with Hoppes#9.  So, look for a better barrel cleaning solvent also helps you and the rifle/pistol.  Oh, BTE copper remover should be used with nylon cleaning brushes as it will eat up your brass/bronze brush (copper remover).

You pull, not push.

Take any cleaning rod, put the end on the floor and push, what happens?  The rod bows.  Inside the barrel the bow will be stopped by the rod rubbing on the inside of the barrel.

As to pulling toward the chamber or toward the muzzle, that doesn't matter.

For most cleaning a bore snake is hard to beat. It goes one way, no issues with that.

For more intensive cleaning with rods, I push it through, remove the first patch or two before withdrawing the rod.
Depending on whether you can actually fix a patch so that it is firmly held, pulling it back after the first couple of patches (to remove the 'grit') probably actually helps.

As mentioned, there is a lot of folklore surrounding cleaning the bore.
As with everything else: Don't be stupid, use appropriate tools and don't go wild.