I'm running a simulation. From what I can see, the clock signals carry current. When P3 turns on, it connects the two lower capacitors directly in series, so that one discharges into the other. (They are joined by a common node at the Ø1 clock sources.) It looks as though it's wasteful, however it does not dissipate watts. It's a method to transfer charge.
P6 does a similar function with the upper two capacitors.
It is convenient to use Pmos because the body diodes perform a chief role at certain times in the cycle. It would not be so easy to do the same using Nmos.
The alternating clock signals provide power to the circuit, similar to applying AC.
The clock edges are unaligned, possibly because momentary shoot-through might occur if they were to coincide.