I use the lines on the mat to line up the fabric and for measuring fabric length and width. If squaring the ends I cut with the fabric to the left but if I'm cutting fabric for piecing I measure with my ruler and the main fabric to my right. It just seems natural to me that way. Plus I think I was taught that way?
I am mainly a right-handed person although I have always done sport related things left-handed. Don't know if this is why I cut fabric the way I do.