The PE6000 and the LM6000 are two shaft machines, where there is a low speed shaft on which sits the low pressure compressor and low pressure turbine (the hot and cold ends), and a high speed shaft on which sits the high pressure compressor and high pressure turbine. The two shafts are concentrically located, with the high speed shaft being on the outside. The low speed shaft is where the generator is coupled, and can be done on either the hot or cold end.

You're right that the core doesn't spin at synchronous speed but the LP shaft does. It's optimized for 3600rpm, but could run at other speeds... the machine just isn't designed for it. The LM6000 only uses a gearbox for 50Hz units while 60Hz don't need it.