First of all, it is absolutely true that any kind of elective surgery can be safely performed during any phase of the menstrual cycle. In fact, often the psychological stress or pre-operative "butterflies" can often send a woman into her cycle prematurely--all surgeons who have been operating more than a few years have seen this, since absolute predictability is impossible!
But, as physicians, we also understand that the female body undergoes significant and profound physiologic changes with each menstrual cycle in order to prepare for the possibility for fertilization of an ovum and resultant pregnancy. These changes affect the uterus primarily, but also the breasts, and to lesser degrees, the rest of the woman's body. Just ask a dozen women or so--they'll tell you! Obviously, some women notice these effects more than others, and surgeons who operate on lots of women start to notice these effects as well.
Of course, since women choose their surgical dates on the basis of personal schedule, work requirements, time off requests, help at home, children's schedules, spousal or partner availability, etc., we operate on women at all phases of their menstrual cycle, and we do it safely and effectively. So that should put to rest the "urban legend" that is is "unsafe" or that there is potentially serious risk of bleeding if operations are performed during a menstrual cycle.
BUT . . .
It would be equally incorrect to state that there is NO identifiable difference in women at various phases of their cycle. Many surgeons, myself included, have operated on enough women to be aware of these differences, and I recommend, all things being equal, that if possible, a woman schedules her surgery a week AFTER her period rather than the week BEFORE. At least with breast surgery, and perhaps with other operative areas as well, there is less bleeding, more easy coagulability, and less "boggy" tissues in the operating room, as well as less bacteria in the breast ducts, less tenderness, bruising, and swelling post-op, and as a result of these factors, perhaps less likelihood of capsular contracture. I would cheerfully admit that this is not a statistically significant, double-blinded, peer-reviewed scientific fact, but something that has become clearly evident to me in over 25 years of breast surgery in thousands of patients.
So, if possible, why not? And if not possible, I know to be just a little bit more cautious, and a tiny bit longer on the vessel coagulation, and a smidgen more restrictive on my patient activity recommendations. That's the art that goes along with the science, or perhaps the experience that goes along with the skill!