There is something to different sides of the brain "thinking" differently. Look up a copy of Drawing on the Right Side of the Brain. It's an art book, but it's written to help the non-artsy let go of the supposedly more analytic left side of the brain and reproduce what they see through the more creative right side. Very interesting book, with lots of comparison exercises to show how each "brain" sees and reproduces the world in drawing.
Supposedly, also in people, the stereotypes associated with "handedness" are attributable to the sides of the brain and that each side manages the opposite side of the body. So, southpaws are right-brain dominant, so more artistic, creative, and less analytic and organized, and vice-versa for the left-brain dominant right handed person, goes the theory.
As far as horses being more comfortable with things on the left vs. the right side of their body, I believe that that's purely a result of time spent on a given side.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown