I grew up with shotguns, but all I knew was to hold on for dear life and disregard any teeth that get rattled. I've since learned of the push pull method, pushing with forearm and pulling with firing hand, kind of like pulling the gun apart, to mitigate recoil.
My question is: what about the pull push method? The same as above but opposing the effort exerted on the gun. Squeezing the front into the back instead of trying to pull it apart. That seems to me to be less taxing on the arms. I'm sure I'm not the first to think of this, so I just wanted to hear if that is helpful at all.

I just use my firing hand to pull the stock tight into the shoulder and the support hand puts some rearward pressure on the pump so as soon as the gun fires, my hand, along with recoil will automatically open the action and all I have to think of is pushing it closed again. Lean into the gun some, but keep my upper body fairly relaxed to roll with the recoil if I'm firing something spicy.

Pushing with your support hand keeps your balance forward and off your back foot.

Use the support (left) hand to send the recoil down through your left foot.

“Pushing” with the trigger hand will tend to take the butt off the shoulder which will exacerbate felt recoil.

but absolutely experiment with different hold techniques. The BEST one is what works for YOU.

Also, since you’re going down this path, don’t ignore the role a proper fitting stock plays in felt recoil mitigation. Length of pull being the most critical measurement but comb profile and other dimensions contributing as well.

People who talk about a gun/load “kicking like a mule” are typically doing it wrong. My 14yo, 85 pound daughter could shoot heavy turkey loads all day long with the gun that actually fit her frame.

I'll sometimes do pull-pull on soft shooting gas semiautos that have a pistol grip.