CPU-Z says my PC3-10600 1333MHz RAM is actually PC3-10700 667MHz. Do I have 1333 or 667 MHz RAM?

You have 1333 DDR (Double Data Rate) RAM.

The RAM itself runs at 666MHz, but because it is "double pumped" its equivalent data rate is twice that, hence 1333MHz.

You do indeed have PC3-10600. CPU-Z shows only the half of it because DDR RAM like the names says - Double Data Rate - can transfer twice as much data on one I/O bus clock, than the actual IO frequency which CPUZ shows.

If you want to dig deeper in this topic check this link, it describes it really well.

Actually its not 10600 but 10700 as the OP says. and it runs at 1334 (2x667) to be exact. Sisoft Sandra measures the same as the description.