I've discovered that the neck on my EBMM guitar has uneven relief on treble and bass sides. On the low E side, there is very slight backbow, while on the high E side of the neck has an almost unnoticeable frontbow (the frets are a bit closer to the strings in the 10th fret area).
Can this be the result of the climate (right now it's really cold and dry)? What should I do?

Thanks in advance!
Are you sure it's bowing and not just the action of the strings giving that appearance? If one side is bowing in the opposite direction of the other, your neck sounds warped and most likely ruined.
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Are you sure it's bowing and not just the action of the strings giving that appearance?

i think this would be worth checking.

try measuring the relief (ideally with feeler gauges) at both sides of the fingerboard and see if you get the same measurement on each side.
Rig Winter 2017:

Fender Jazzmaster/Yamaha SG1000
Boss TU-3, DS-2, CS-3, EHX small stone, Danelectro delay
Laney VC30-112 with G12H30 speaker, or Session Rockette 30 for smaller gigs
Elixir Nanoweb 11-49 strings, Dunlop Jazz III XL picks
Shure SM57 mic in front of the amp
in my experience there will always be slightly more bend on the treble side, every guitar i have come in contact with was like this. if it's extreme however, it could a be problem. it could also be the strings you use. i remember putting a set on where the gauges had a crappy ratio and it made it do that.
Well, when I press both E strings at the 1st and the 17th frets, there's a bit less space between the high E string and the frets. Both small portions of the strings ring out, though.
But how does it play? If you don't have a problem with that, then I wouldn't worry about it.

I got a baritone recently that had a very warped and twisted neck because it had been stored for a long time, cats had peed on it and obviously the string tension was enough to warp it.

The only fix for mine (it was so bad) was to remove the fretboard and straighten everything up with steam and heat, dry it out and glue it back together. I even ended up breaking the neck joint, doing some sanding and regluing that too. Lots of work but in the end, it came out spot-on.