Fitness

Asymmetry can indicate imbalance in some way. Could be worth evaluating how you handle weight in each arm, maybe record clips if you can?

I've developed pretty bad callouses over the years, though honestly they're handy for certain types of work, particularly deadlift and barbell row. Helps for specific instruments like the guitar or bass, and tasks like sewing.

Climbers tend to file theirs down, I gather it's detrimental to their grip. You may find this helpful though personally I find them valuable for weightlifting. If this is causing acute discomfort you may want to look into using gloves or chalk?

I'm of the opinion that gloves are a solution for avoiding or limiting calluses, but with the asterisks that: 1) it can also be a (minor) crutch if not also developing your grip strength in tandem, and 2) that gloves must be properly fitting or else it might be even worse than no gloves at all.

If the calluses seem to relate only to isolation exercises, then #1 is less of an issue. But if they appear due to compound exercises, then that's a clue to improve your grip, as a callus ostensibly is due to uneven pressure around your hands.

For #2, ill-fitting gloves might grip the implement or barbell or dumbbell, but if the glove slides against your palm, then it'll bunch up and that virtually guarantees uneven pressure. You might also consider fingerless gloves, so that there's material only where it's needed to avoid palm calluses.

Make sure you place the bar/handle along the crease where your palm meets your fingers. That way your skin doesn't get pinched, and callousing is reduced.

A proper grip shouldn't be painful, unless you're doing hook grip for deadlifts.

Are you just starting out? What sort of exercises does the problem occur for?