According to the comment, what this question is really asking us to address is something completely different from “How does the twin (clock) paradox (in SR) really work?”
What the comment asks us to explain is as follows: “A clock flies around the equator eastwards, it ages slightly less. A clock flies around the equator westwards, it ages slightly more. Than a clock which stayed, at home.”
The explanation for this (in the context of either Special OR General Relativity) is that the alleged effect exists only for motion relative to the Earth’s surface (which is already rotating). So if I stand still on the Earth’s surface I will age more slowly than a twin who stands still relative to the Earth’s centre (which entails flying Westwards at a rate of about 1000mph) and more quickly than a twin who flies Eastwards and so is actually moving more quickly relative to the Earth’s centre.
Note: As with the usual out-and-back twin “paradox” what makes the travelers different from the stay-at-homes is that they are not in fixed inertial frames but are accelerated. And in the case of this scenario the twin flying West at 1000mph is basically staying stationary wrt the Earth’s axis which (if we neglect the curvature of the Earth’s orbit) is pretty close to being in an inertial frame. But the one standing still is actually rotating with the Earth and so is accelerating (centripetally) towards the axis. And the one flying East is actually accelerating even more. According to SR, an observer who is accelerated figures that stationary clocks towards which she is accelerating are speeded up by an amount that more than counterbalances the fact that if the speed was fixed she would figure that they were slowed down. And if you do all the calculations it turns out that when they get back together they all agree on their relative ages (with Eastflier younger than Standstill and Standstill younger than poor old Westflier(who is actually the only one who is really standing still)).