Every flash bracket I've ever seen is meant to be held with the left hand. The right hand operates the camera. Wrap the extra flash cord around the bracket. Some models are more vertically friendly than others, but you just have to get used to it. Some flash brackets will flip the flash back to the top even when turned for a vertical shot.
EDIT: The real problem is that most flash brackets are designed the same way today as they were 50 years ago. It wasn't until almost the 1990s that camera bodies began being designed more ergonomically with a grip projection near the shutter and vertical grips. For some reason no one has redesigned the flash bracket to function better with the evolution of the camera body. If I use my vert grip on my flash bracket and turn the camera vert with the flash on top (I have a flip bracket) the shutter button ends up at the bottom, by the base of my palm. I take my grip off, but I don't know what you're supposed to do if your have one of those fancy pro-models where the vert grip isn't removable.