Watching Britain's Young Musician of the Year, I noticed that a French horn player used her left hand to operate the valves. Most (all?) other brass instruments are configured for right hand operation. Why would the French horn be opposite?
It's to do with the origin of the instrument. In baroque times the horn had no valves. The way the player could alter pitch was with the lips and also with the right hand, which was pushed inside the bell. The left hand only helped hold the horn. The right hand could push further inside and cause the pitch to change, but with a much more brassy sound.
Once valves were introduce in the 19th century, they left the general set up the same, with the right hand inside the bell, and therefore the valves were operated with the left hand.
Just history. The dexterity required for hand-stopping isn't enormous.
I started on trumpet, had a try at horn and ended up a trombonist. When I first picked up the horn I barely noticed I was fingering with the 'wrong' hand.