There's a manufacturer making what they call either low or reduced recoil, yet it's still full power. How can a "full power" load have reduced recoil?

A 38 special can be full power and still be a reduced load if it does not enter the +P velocity.

Just depends on how you look at it.

It is very easy if you are writing copy for advertising; not so easy if you are sitting at the loading bench. Which "power" do you mean? Energy? Momentum? Some other value?

Power in competition is defined as "Power factor" which is momentum in odds units (speed times bullet weight).
Since recoil impulse and momentum are not the same, many use a heavy bullet, fast powder, and slow speed to make "Power Factor" with loads that have a little less calculated recoil and also a slower exit of the bullet, which spreads the recoil and reduces the "recoil impulse" or sharpness of the kick.
PERCEIVED recoil is highly subjective and varies widely from person to person.
Here's a handy recoil calculator:
Recoil Calculator

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Science plus Art

Who told you it is FULL power? And what do they consider FULL?

The Laws of Physics do not get broken easily.

For example the 357Magnum Short Barrel 135 grain load that is out there by one of the big manufacturers is touted as reduced recoil. If you look at the ballistics, it is hardly a 38+P+ in performance, but it is in a 357 case and technically more powerful than the SAAMI spec for 38. The same projectile in the non SB, non reduced recoil version gets several hundred more FPS and that is still not a true FULL power 357 Magnum

"For every action there is an equal but opposite reaction", some guy named Newton....

Wayne Newton is still alive?

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Minimize the variables