I stand southpaw, with my less flexible right foot in front. To throw a roundhouse kick, my Kyokushin training makes it feel natural for me to plant my left foot solid on the ground to kick with the right. Maybe occasionally I will slightly rotate the standing (non-kicking) foot, if I am going for a Thai style roundhouse. Is it OK to plant the standing foot, or should I rotate the standing foot to throw a roundhouse kick?
In general, the more power you want to put in a roundhouse kick, and the more you want it to penetrate the target, the more you should pivot that supporting foot (and possibly start by stepping to the side of the kick). The reason why is fairly obvious in that you need to be able to deliver the full power of the kick close to the center of your body and, without pivoting the support leg, that means you're essentially either not using your hips as much (a common situation for snap roundhouses in Tae Kwan Do, especially to the head) or your supporting leg is essentially being forced to rotate heavily in the other direction (since the rest of your body is rotating).