Some of the really heavily bred barrel horses I have noticed are excellent english horses, because they are bred to be free moving. This is just what I have noticed, and it was been pegged true in my experience.
My mare is cow bred, and she is a catty little barrel horse but she isn't a good english horse. She can do hunt seat okay, and equitation, but she isn't excellent in them.
If you want to do both, switch tack and bits. I do this with horses that do multiples. For example, Selena is my main girl right now, and she runs barrels in a jr. cowhorse bit and short short short barrel reins. Then she does reining and pattern classes in a long shank billy allen bit. She's a lot hotter in the jr. cowhorse than she is in the billy allen, because she knows what that means.
But, on that same note, a horse doesn't care what saddle is on their back, so long as you ride them well. It doesn't need to be a big deal to them.
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.