Virtually every instrument I can think of is asymmetric, favouring the right hand for the more dexterous actions. I know of guitarists who play left handed on an inverted instrument but I've never seen any others. Perhaps there is a historical conspiracy to keep us devil-handed music lovers off the concert stage? In particular, is there such a thing as a left handed piano (or keyboard instrument of any kind)?
The Great Highland Bagpipe can be played either from the left (conventional) or from the right (the drones would need to be rearranged and the bag cut slightly differently). I always heard the less-common alternative referred to as Lewis pipers, but when I Googled it, that term did not show anything with that arrangement. Switching is it easily achieved for left-handed players if they are more comfortable flipping the instrument. That said, I can't see an advantage to having the bag under the left or right arm. The conventional, left-arm approach, requires the left hand to play the "upper hand" of the chanter, and reversing the bag reverses the hands so the right hand would play the upper notes. Since both hands are required to do things of equal dexterity and strength, there seems to me to be no advantage one way or the other.
I once had a left-handed student who complained that he was having a hard time learning to play the conventional - presumably right-handed - way. I suggested he switch, and I did that to see how it would work, I found that with a little mental adjustment, I could play it either way. And either way he was still having problems, so we both decided that more practice rather than switching an arbitrary arrangement of hands was probably the answer.
I'm a right-handed musician. I play flute and violin.
Both of these favour left-handed players. With the flute it's quite subtle, but a lot of the manual dexterity (hah!) required comes from the left hand, which tends to be more involved in higher notes - particularly in the third octave.
The violin is absolutely a left-handed instrument. While bow control is the heart of the sound, it uses more massive, less intricate movements than the absolute precision needed with the left hand.
Some violinists do play the instrument 'left-handed' - with the violin resting on the right shoulder. This is usually done for reasons of disability - commonly an injured left shoulder.
As I don't play them, I can't speak for other instruments, but looking around an orchestra it seems to me that the majority of instruments require at least equal dexterity from both hands, and many do seem to favour lefties. The entire brass section, for a start.