From what I understand from past applicants, many candidates will receive interview offers with conflicting dates, especially Saturdays. For example, 9 of the programs I applied to have January 5th listed as one of several or their only interview date. Depending on how many places you applied, you are quite likely to run into this situation again before interview season is over. If you are serious about both UCLA & Mayo, I would ask about the other possible UCLA dates listed on the website, keeping in mind that you may thereby let them know they are not your top priority. However, all coordinators are aware of what we applicants go through trying to schedule everything, and most are understanding.
With this in mind, if you have a few absolute favorite programs, it's worth keeping their interview dates (if you know them) in mind and trying to keep those dates free if possible when scheduling other interviews that offer a choice of dates.
Since there's a good chance you will have a conflict that cannot be resolved at some point and you will have to cancel one interview in favor of the other, it's also a good idea to have a rough idea of how interested you are in each program - not a ranking, but more like most interested, somewhat interested, and not very interested but would train there in order to become an orthopod. Everyone of us is stressed, and it will suck if/when we have to cancel interviews, but it's also a good problem to have since it means you have several/a lot of interviews. Hope that helps.