You can go to both but what I do and see most often is the right one....
But like I said it doesn't matter which one you go to first which everyone works for you and your horse.
I see that Fallon Taylor takes her horse Baby Flo on either barrel. I've seen them win going left first and I've seen then win going right first.
My heading mare likes to go right first because:
A) she sucks at turning right
B) as a heading horse, her left turns are naturally stronger, so why fight it?
Now Squiggy, she turns right better, so we go left first, but I'm sure I could go either if I wanted.
Selena runs to the right first. I've been thinking about taking her to the left every now and then though, just because. She's a strong turner both directions.
I start all mine to the right first but if I feel they are struggling or would work better I switch them. BB and Scrat go to the left first and all the others go to the right first.
I recently switched Scrat to the left because I was fighting him with 2 barrels and so I was told to switch at a clinic and everything just was as smooth as ice since then.