Hmmm. swapping tires is tricky depending on how hard you drive. If you are tracking your car then here is what I've been told by experienced racers that I trust and also experienced myself.
Never swap front/rear tires (ok, so different size on Cayman so you're not going to be doing that but for example on my MX-5, they are the same size front/rear and this would be a no-no. Front/rear tires wear very differently so its not a good idea at all to swap them.
You can swap wheels left to right by removing the wheels and just putting the left wheel as is on the right and vice versa. You can only do this in the dry on most tires as most thread is designed to channel water one way and not the other. Doing this, you should still build up speed easily over 10 minutes or so to make sure everything is ok. Should be fine though. This is useful is you drive a lot of circuits where the corners are mostly in one direction and one side gets worn before the other. I rotate my two front tires often as otherwise my left tire's outside shoulder just wastes away in no time.
You should not switch the tires so that inside becomes outside and vice versa. I know its tempting, I've done it myself but its a bad move. Cornering hard eats away at the outside of the tires and they become diagonal over time. Camber also effect this. Tires also develop their best grip when they develop this diagonalness as the rubber touching the surface is longer than a straight edge. If you rotate tires inside/outside, you'll basically be riding peaks on the rubber so grip will be very much compromised. If you don't corner hard, you may never notice, but when driving hard, you basically have very reduced grip. Over time, the tire will grip again but by the time it does, you really need new tires anyway. I recommend you don't do this at all.
While I think of it, tires are best replaced as a set, you can replace only front two or only back two but never just the one if you are tracking your car hard. You can get away with it, but you need to be cautious. Also for example, if rear left tire has 4mm and you get a new right tires that has 8mm, the tire rotation is different as the diameter is different so it can cause the
lsd to wear faster over time.
Instead think of it like this... your car is connected to the road by the 4 tires that your car rides on. You accelerate, brake and change direction through only those. Hence they are very important! Good tires are the best modification you can do on your car.
Another point is that different tires have different advantages/disadvantages to suit your needs. Some wear longer than others. Some grip better than others and some let go grip better than others. Use this time to learn more by trying out some other tires. My current favourite tire is the Bridgestone RE11. Not sure if they are available in the US yet or not.
Anyway, enjoy your track days and enjoy your Cayman!
