No one else seems to have said this yet, so here goes. My initial bodywork training was in Thai massage, and I was taught to start *in general* on the left for women and on the right for men,due to a natural energetic affinity between each side and gender. If you study eastern philosophies and medicine, you'll notice the correspondences yin/yang, female/male, left/right, dark/light, etc.
So, as a habit, I generally follow my sessions through in this way. Example: male client - start prone right back, left back, left leg, right leg, supine right leg, left leg, abdominal, left arm, right arm, neck, head, face. Sorry for the lack of diagram
However, if a client presents with a complaint on one side over the other, I tend to start work on that side just to bring them a sense of peace that I am addressing their concerns, and also to give me enough time to assess and work out what I can in that area. Personal experience too. I know that I hate it when I go for a massage and complain of my left levator hurting, and that's not at the top of the list of areas the therapist touches.
Original question - I have no idea why anyone would be taught to always start on one side of the client. It seems rather random to me. Or perhaps misunderstood. Perhaps the teacher/therapist meant that to be a guideline as one is learning massage therapy? Don't know. The energetic correspondences have always made sense to me.