A gas turbine engine comprises a main compressor section having a high pressure compressor with a downstream discharge, and more upstream locations. A turbine section has a high pressure turbine. A tap taps air from at least one of the more upstream locations in the compressor section, passes the tapped air through a heat exchanger and then to a cooling compressor. The cooling compressor compresses air downstream of the heat exchanger, and delivers air into the high pressure turbine. The cooling compressor is connected to be driven with at least one rotor in the main compressor section. A source of pressurized air is selectively sent to the cooling compressor to drive a rotor of the cooling compressor to rotate, and to in turn drive the at least one rotor of the main compressor section at start-up of the gas turbine engine. An intercooling system is also disclosed.